Kaggle Competition

Preparation

Load Data

First, let’s set up the work directory and take a glance of the raw analysis data.

setwd("/Users/arvinzhang/Desktop/APAN5200_Kitty/Kaggle/Kaggle_2")
analysis_raw <- read_csv("analysisData.csv")

## Warning: One or more parsing issues, see `problems()` for details

## Rows: 41330 Columns: 91

## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr  (35): name, summary, space, description, neighborhood_overview, notes, ...
## dbl  (42): id, host_listings_count, host_total_listings_count, zipcode, acco...
## lgl  (11): host_is_superhost, host_has_profile_pic, host_identity_verified, ...
## date  (3): host_since, first_review, last_review

## 
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.

head(analysis_raw)

## # A tibble: 6 × 91
##       id name   summary  space description neighborhood_ov… notes transit access
##    <dbl> <chr>  <chr>    <chr> <chr>       <chr>            <chr> <chr>   <chr> 
## 1 914965 <BRAN… Trendy,… Ther… Trendy,  a… <NA>             <NA>  The J/… <NA>  
## 2 262763 <Broo… Full pa… <NA>  Full parlo… <NA>             <NA>  <NA>    There…
## 3 279053 <Posh… Brand n… The … Brand new … Short and easy … Whil… Severa… Acces…
## 4 385319 <COMF… *Beauti… The … *Beautiful… There are many … I ha… Parkin… Bathr…
## 5 476558 <Sunn… A quick… <NA>  A quick wa… <NA>             This… It is … <NA>  
## 6 522365 <Cozy… Nice an… <NA>  Nice and c… <NA>             <NA>  Good r… <NA>  
## # … with 82 more variables: interaction <chr>, house_rules <chr>,
## #   host_name <chr>, host_since <date>, host_location <chr>, host_about <chr>,
## #   host_response_time <chr>, host_response_rate <chr>,
## #   host_acceptance_rate <chr>, host_is_superhost <lgl>,
## #   host_neighbourhood <chr>, host_listings_count <dbl>,
## #   host_total_listings_count <dbl>, host_verifications <chr>,
## #   host_has_profile_pic <lgl>, host_identity_verified <lgl>, street <chr>, …

Locate Missing Data

First, let’s see the cleanliness of the data set. We can see that there are several variables have large number of missing values, we need to filter them out as it is unnecessary to the analysis.

miss_var <- miss_var_summary(analysis_raw); miss_var

## # A tibble: 91 × 3
##    variable           n_miss pct_miss
##    <chr>               <int>    <dbl>
##  1 jurisdiction_names  41330    100  
##  2 license             41317    100. 
##  3 monthly_price       41126     99.5
##  4 square_feet         40972     99.1
##  5 weekly_price        38157     92.3
##  6 notes               22965     55.6
##  7 access              17074     41.3
##  8 host_about          15222     36.8
##  9 interaction         14760     35.7
## 10 house_rules         14633     35.4
## # … with 81 more rows

miss_var %>%
  filter(pct_miss > 0) %>%
  ggplot(aes(x = reorder(variable, pct_miss), y = pct_miss)) +
  geom_col(fill = "skyblue") +
  coord_flip() +
  labs(title = "Proportions of Missing Values by Variables",
       x = "Proportions of Missing Values",
       y = "Variables")

Filter Missing Data less than 25%

Filter for variables whose pct_miss is less than 25%, leaving 78 variables.

var_keep <- miss_var %>%
  filter(pct_miss < 25)
datatable(var_keep)

Find Other Type of Missing Data

See if there are other types of missing data.

Done, omit these 9 missing variables first to build models.

miss_na_var <- analysis_raw %>%
  select(var_keep$variable) %>%
  miss_scan_count(search = list("N/A", "na")) %>%
  arrange(desc(n)) %>%
  filter(n > 0)
datatable(miss_na_var)

1st Round of Data Cleaning

Omit missing variables

Now, we would need to filter variables with many missing values. Some steps weren’t showed as I keep using function skim() to check the features and distributions of each variables.

According to general knowledge, we can manually clear irrelevant variables.

Also, we can see some variables are character but actually are categorical factors such as neighbourhood_group_cleansed, property_type, room_type, bed_type, and cancellation_policy, we need to convert them into factors so that we can conduct further analysis.

By looking at the distribution of some variables such as has_availability, requires_license, is_business_travel_ready,we can see they only contain single value, meaning they are unimportant to the analysis, so let’s omit them.

Moreover, there are some rows with prices equal to 0, which makes no sense, thus we need to clear them out.

At last, we need to fill missing values, we use function fill() for variables with integer and logical values such as beds, host_is_superhost, host_total_listings_count, host_has_profile_pic, and host_identity_verified, and use mean values to fill missing values for continuous factors like reviews_per_month, and cleaning_fee.

analysis <- analysis_raw %>%
  # filter variables with many missing values
  select(var_keep$variable) %>%
  select(!miss_na_var$Variable) %>%
  # omit unimportant variables according to general knowledge
  select(-host_listings_count, -market, -id, 
         -zipcode, -number_of_reviews, - state, -country_code, -country,
         -calendar_updated, -host_verifications, -require_guest_profile_picture, 
         -require_guest_phone_verification) %>%
  # change character to factor
  mutate(neighbourhood_group_cleansed = as.factor(neighbourhood_group_cleansed),
         property_type = as.factor(property_type),
         room_type = as.factor(room_type),
         bed_type = as.factor(bed_type),
         cancellation_policy = as.factor(cancellation_policy)) %>%
  # omit single value variables
  select(-has_availability, -requires_license, -is_business_travel_ready) %>%
  # clear rows with price == 0
  filter(price != 0) %>%
  # fill missing values in host_is_superhost, host_has_profile_pic, host_identity_verified, and
  # host_total_listing_count
  fill(host_is_superhost, host_total_listings_count, host_has_profile_pic, host_identity_verified, 
       first_review, last_review, host_since) %>%
  # place price column to the first place
  select(price, everything())

# change first_review
analysis$first_review = as.numeric(floor(difftime("2020-12-31", analysis$first_review, units = "days")))
# change last_review
analysis$last_review = as.numeric(floor(difftime("2020-12-31", analysis$last_review, units = "days")))
# change host_since
analysis$host_since = as.numeric(floor(difftime("2020-12-31", analysis$host_since, units = "days")))

# fill missing values in beds with ceiling(mean(beds))
analysis$beds[is.na(analysis$beds)] = ceiling(mean(analysis$beds, na.rm = T))
# fill missing values in cleaning_fee with avg. cleaning_fee
analysis$cleaning_fee[is.na(analysis$cleaning_fee)] = mean(analysis$beds, na.rm = T)
# fill rows with cleaning_fee = 0 with mean(cleaning_fee)
analysis[analysis$cleaning_fee == 0, c("cleaning_fee")] = mean(analysis$cleaning_fee)
# fill missing values in reviews_per_month with avg. reviews_per_month
analysis$reviews_per_month[is.na(analysis$reviews_per_month)] = mean(analysis$reviews_per_month, na.rm = T)

# check if there are still missing values
datatable(miss_var_summary(analysis))

Clean High-dimensional Variables

Explore Property Type

Notice that property_type has 33 unique values, it may be too much for modeling, let’s see what’s going on.

analysis %>%
  group_by(property_type) %>%
  count() %>%
  arrange(desc(n))

## # A tibble: 33 × 2
## # Groups:   property_type [33]
##    property_type          n
##    <fct>              <int>
##  1 Apartment          32212
##  2 House               3620
##  3 Townhouse           1450
##  4 Condominium         1393
##  5 Loft                1226
##  6 Guest suite          367
##  7 Serviced apartment   235
##  8 Boutique hotel       196
##  9 Hotel                175
## 10 Guesthouse            71
## # … with 23 more rows

analysis %>%
  group_by(property_type) %>%
  count() %>%
  ggplot(aes(x = reorder(property_type, n), y = n)) +
  geom_col() +
  coord_flip() +
  labs(title = "Property Types of NY Airbnb Rentals", 
       x = "Number of Rentals", 
       y = "Property Type")

Reduce Dimensionality of Variables

Obviously, there are too many categories in property_type, we need to collapse small size factors into Others, and categorize them into specific categories.

Also, cancellation_policy has 6 categories and the size of strict factor is pretty small, so we can incorporate them into strict category. Noticed cancellation_policy is actually ordinal variable, so we can convert it into a dummy variable.

analysis <- analysis %>%
  # collapse property_type factors
  mutate(
    property_type = fct_collapse(property_type, 
                                 "Apartment" = c("Apartment", "Serviced apartment"),
                                 "Condominium" = "Condominium",
                                 "House" = c("Bungalow", "House", "Guesthouse", "Hostel", 
                                             "Tiny house", "Townhouse", "Treehouse", 
                                             "Dome house", "Villa"),
                                 "Loft" = "Loft",
                                 "Hotel" = c("Hotel", "Bed and breakfast", "Boutique hotel", 
                                             "Guest suite", "Resort"))
    ) %>%
  mutate(property_type = fct_other(property_type, keep = c("Apartment", "Condominium", "House", 
                                                           "Loft", "Hotel"))) %>%
  # collapse cancellation_policy
  mutate(cancellation_policy = fct_collapse(cancellation_policy, 
                                       "strict" = c("strict", "super_strict_30", "super_strict_60",
                                                    "strict_14_with_grace_period"))) %>%
  # transform cancellation_policy into ordinal variables
  mutate(cancellation_policy = case_when(
    cancellation_policy == "flexible" ~ 3,
    cancellation_policy == "moderate" ~ 2,
    cancellation_policy == "strict" ~ 1
  ))

Take a Look

So far, we finished the 1st round of data cleaning. Congrats!! At last, we have 48 variables in analysis data set. Perfect!

# see the structure of analysis
skim(analysis)

Data summary

Name	analysis
Number of rows	41322
Number of columns	48
_______________________
Column type frequency:
character	1
factor	4
logical	5
numeric	38
________________________
Group variables	None

Variable type: character

skim_variable	n_missing	complete_rate	min	max	empty	n_unique	whitespace
amenities	0	1	1	1500	0	38901	0

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
neighbourhood_group_cleansed	0	1	FALSE	5	Man: 17695, Bro: 17035, Que: 5194, Bro: 1056
property_type	0	1	FALSE	6	Apa: 32447, Hou: 5290, Con: 1393, Lof: 1226
room_type	0	1	FALSE	4	Ent: 21512, Pri: 18464, Sha: 1101, Hot: 245
bed_type	0	1	FALSE	5	Rea: 40759, Fut: 221, Pul: 189, Air: 105

Variable type: logical

skim_variable	n_missing	complete_rate	mean	count
host_is_superhost	0	1	0.22	FAL: 32277, TRU: 9045
host_has_profile_pic	0	1	1.00	TRU: 41208, FAL: 114
host_identity_verified	0	1	0.46	FAL: 22217, TRU: 19105
is_location_exact	0	1	0.83	TRU: 34141, FAL: 7181
instant_bookable	0	1	0.40	FAL: 24879, TRU: 16443

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
price	0	1	137.04	111.06	10.00	66.00	100.00	170	9.990000e+02	▇▁▁▁▁
cleaning_fee	0	1	58.17	52.33	1.58	20.00	50.00	80	6.000000e+02	▇▁▁▁▁
beds	0	1	1.58	1.15	0.00	1.00	1.00	2	2.600000e+01	▇▁▁▁▁
host_since	0	1	1987.05	915.91	209.00	1267.00	2006.00	2683	4.509000e+03	▅▆▇▅▁
host_total_listings_count	0	1	8.79	72.53	0.00	1.00	1.00	2	2.345000e+03	▇▁▁▁▁
first_review	0	1	1185.33	712.32	206.00	594.00	999.00	1640	4.329000e+03	▇▅▂▁▁
last_review	0	1	628.51	437.25	205.00	333.00	455.00	729	3.521000e+03	▇▁▁▁▁
reviews_per_month	0	1	1.35	1.59	0.01	0.22	0.76	2	3.371000e+01	▇▁▁▁▁
accommodates	0	1	2.93	1.91	1.00	2.00	2.00	4	1.800000e+01	▇▁▁▁▁
bathrooms	0	1	1.14	0.41	0.00	1.00	1.00	1	7.000000e+00	▇▁▁▁▁
bedrooms	0	1	1.19	0.75	0.00	1.00	1.00	1	2.100000e+01	▇▁▁▁▁
guests_included	0	1	1.60	1.23	1.00	1.00	1.00	2	1.600000e+01	▇▁▁▁▁
extra_people	0	1	16.19	24.94	0.00	0.00	10.00	25	3.000000e+02	▇▁▁▁▁
minimum_nights	0	1	5.99	17.00	1.00	1.00	2.00	4	1.250000e+03	▇▁▁▁▁
maximum_nights	0	1	1552.74	139136.91	1.00	29.00	365.00	1125	2.000000e+07	▇▁▁▁▁
minimum_minimum_nights	0	1	5.63	15.72	1.00	1.00	2.00	4	1.250000e+03	▇▁▁▁▁
maximum_minimum_nights	0	1	7.68	45.86	1.00	2.00	3.00	4	4.513000e+03	▇▁▁▁▁
minimum_maximum_nights	0	1	365404.84	27948708.24	1.00	30.00	1125.00	1125	2.147484e+09	▇▁▁▁▁
maximum_maximum_nights	0	1	365407.95	27948708.19	1.00	30.00	1125.00	1125	2.147484e+09	▇▁▁▁▁
minimum_nights_avg_ntm	0	1	6.96	29.80	1.00	1.30	2.00	4	1.587100e+03	▇▁▁▁▁
maximum_nights_avg_ntm	0	1	365405.91	27948708.22	1.00	30.00	1125.00	1125	2.147484e+09	▇▁▁▁▁
availability_30	0	1	11.20	11.76	0.00	0.00	7.00	23	3.000000e+01	▇▂▂▂▃
availability_60	0	1	23.17	23.07	0.00	0.00	17.00	45	6.000000e+01	▇▂▂▂▃
availability_90	0	1	35.88	34.69	0.00	0.00	29.00	71	9.000000e+01	▇▂▂▂▅
availability_365	0	1	126.16	132.62	0.00	0.00	79.00	245	3.650000e+02	▇▂▂▁▃
number_of_reviews_ltm	0	1	11.21	16.78	0.00	1.00	4.00	16	3.510000e+02	▇▁▁▁▁
review_scores_rating	0	1	93.61	9.37	20.00	91.00	96.00	100	1.000000e+02	▁▁▁▁▇
review_scores_accuracy	0	1	9.58	0.92	2.00	9.00	10.00	10	1.000000e+01	▁▁▁▁▇
review_scores_cleanliness	0	1	9.25	1.13	2.00	9.00	10.00	10	1.000000e+01	▁▁▁▁▇
review_scores_checkin	0	1	9.71	0.80	2.00	10.00	10.00	10	1.000000e+01	▁▁▁▁▇
review_scores_communication	0	1	9.72	0.82	2.00	10.00	10.00	10	1.000000e+01	▁▁▁▁▇
review_scores_location	0	1	9.58	0.78	2.00	9.00	10.00	10	1.000000e+01	▁▁▁▁▇
review_scores_value	0	1	9.37	0.97	2.00	9.00	10.00	10	1.000000e+01	▁▁▁▁▇
cancellation_policy	0	1	1.71	0.82	1.00	1.00	1.00	2	3.000000e+00	▇▁▃▁▃
calculated_host_listings_count	0	1	5.69	27.66	1.00	1.00	1.00	2	3.270000e+02	▇▁▁▁▁
calculated_host_listings_count_entire_homes	0	1	4.03	26.88	0.00	0.00	1.00	1	3.190000e+02	▇▁▁▁▁
calculated_host_listings_count_private_rooms	0	1	1.44	6.27	0.00	0.00	0.00	1	1.500000e+02	▇▁▁▁▁
calculated_host_listings_count_shared_rooms	0	1	0.14	1.04	0.00	0.00	0.00	0	2.400000e+01	▇▁▁▁▁

Explore Data Analysis

Price Distribution

Let’s see the distribution of price. We can see that price is heavily right-skewed, so we may need to logarithm the price using linear regression.

analysis %>%
  ggplot(aes(price)) +
  geom_histogram(bins = 50)

Factorial Variables

Neighbourhood Grop Cleansed

Apparently, Manhattan has the highest rental price, followed by Brooklyn, and Bronx has the lowest rental price level.

analysis %>%
  ggplot(aes(x = fct_reorder(neighbourhood_group_cleansed, price), log(price))) +
    geom_boxplot() +
    labs(title = "Prices Level in log by Neighbourhood", 
       x = "Area", 
       y = "Price, log")

Property Type

Price vs property_type

It seems Loft and Condominium have highest price level, while House is the lowest.

analysis %>%
  ggplot(aes(x = fct_reorder(property_type, log(price)), log(price))) +
  geom_boxplot() +
  labs(title = "Prices Level in log by Property Type", 
       x = "Property", 
       y = "Price, log")

Distribution of property_type

analysis %>%
  ggplot(aes(x = property_type)) +
  geom_bar() +
  labs(title = "Number of rentals by Property Type", 
       x = "Property Type", 
       y = "Number of Rentals")

Room Type

analysis %>%
  ggplot(aes(x = fct_reorder(room_type, price), log(price))) +
  geom_boxplot() +
  labs(title = "Prices Level in log by Room Type", 
       x = "Room Type", 
       y = "Price, log")

analysis %>%
  ggplot(aes(x = room_type)) +
  geom_bar() +
  labs(title = "Number of Rentals by Room Type", 
       x = "Room Type", 
       y = "Number of Rentals")

Bed Type

Well, almost 99% of bed are real bed, and it would do nothing in predicting the prices. Omit it.

analysis %>%
  ggplot(aes(x = bed_type)) +
  geom_bar() +
  labs(title = "Number of Rentals by Bed Type", 
       x = "Bed Type", 
       y = "Number of Rentals")

Logical Variables

Next we take a look at logical variabels.

There are 99.7% of host_has_profile_pic are TRUE, meaning the variable may not have significant impact on the price. Omit this variable.

analysis %>%
  select_if(is.logical) %>%
  pivot_longer(cols = 1:5, names_to = "predictors", values_to = "values") %>%
  group_by(predictors, values) %>%
  count() %>%
  ungroup() %>%
  ggplot(aes(x = values, y = n)) +
  geom_col() +
  facet_wrap(predictors ~., scales = "free") +
  labs(title = "Distributions Rentals of Factorial Variables",
       x = "Logical Factors",
       y = "Number of Rentals")

Numeric Variables

Cleaning Fee

Let’s first take a look of the relationship between price and cleaning_fee, we can see that it has a correlation between them, indicating that cleaning_fee is a key factor of price.

analysis %>%
  ggplot(aes(cleaning_fee, price)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm", se = F) +
  labs(title = "Price in log by Cleaning Fee",
       x = "Cleaning Fee",
       y = "Price, log")

## `geom_smooth()` using formula 'y ~ x'

Accommodates

Avg. Price with Accommodates

It’s not obvious by merely looking at the distribution of each accommodate with price level. We can cansider discretize accommodates to see if there is a relationship.

analysis %>%
  group_by(accommodates) %>%
  summarize(n = n(),
            mean = round(mean(price), 4)) %>%
  datatable()

Feature Discre.- Accommodates

It’s obvious now that price increases when the accomodates increase, we can keep this variable for later analysis.

We can see that there is only one observation in the group (16,19], so we can later put it in (13,16] if we use this discretized variable acco_cut.

acco_cut <- analysis %>%
  summarize(acco_cut = cut(accommodates, seq(1, 19, 3), include.lowest = T))
analysis %>%
  cbind(acco_cut) %>%
  group_by(acco_cut) %>%
  summarize(n = n(),
            mean_price = mean(price),
            mean_clean = round(mean(cleaning_fee), 4)) %>%
  datatable()

Beds

Distribution of Price by Beds

There is also a correlation between price and beds, we can include it in later modeling.

analysis %>%
  ggplot(aes(beds, log(price))) +
  geom_point() +
  geom_smooth(method = "lm", se = F) +
  labs(title = "Price in log by Beds",
       x = "Beds",
       y = "Price, log")

## `geom_smooth()` using formula 'y ~ x'

Price with Beds

Let’s see if it’s necessary to discretize beds.

analysis %>% 
  group_by(beds) %>%
  summarize(n = n(),
            mean = mean(price)) %>%
  datatable()

Feature Discre. - Beds

As we can see, we can collapse observations whose the number of beds is greater than 12.

beds_cut <- analysis %>%
  summarize(beds_cut = cut(beds, seq(0, 27, 3), include.lowest = T))
analysis %>%
  cbind(beds_cut) %>%
  group_by(beds_cut) %>%
  summarize(n = n(), 
            mean_price = round(mean(price), 4),
            mean_clean = round(mean(cleaning_fee), 4)) %>%
  datatable()

Bedrooms

Distribution of Price by Bedrooms

I will explore the rest numeric variables as I did above.

analysis %>%
  ggplot(aes(bedrooms, log(price))) +
  geom_point() +
  geom_smooth(method = "lm", se = F) +
  labs(title = "Price in log by Bedrooms",
       x = "Bedrooms",
       y = "Price, log")

## `geom_smooth()` using formula 'y ~ x'

Avg. Price with Bedrooms

analysis %>%
  group_by(bedrooms) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Feature Discre. - Bedrooms

bedrooms_cut <- analysis %>%
  summarize(bedrooms_cut = cut(bedrooms, seq(0, 10, 2), include.lowest = T))
analysis %>%
  cbind(bedrooms_cut) %>%
  group_by(bedrooms_cut) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Bathrooms

Distribution of Price by Bathrooms

analysis %>%
  ggplot(aes(bathrooms, log(price))) +
  geom_point() +
  geom_smooth(method = "lm", se = F) +
  labs(title = "Price in log by Bathrooms",
       x = "Bathrooms",
       y = "Price, log")

## `geom_smooth()` using formula 'y ~ x'

Avg. Prics with Bathrooms

analysis %>%
  group_by(bathrooms) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Feature Discre. - Bathrooms

bathrooms_cut <- analysis %>%
  summarize(bathrooms_cut = case_when(
    bathrooms %in% c(0.0, 0.5, 1.0) ~ 1,
    bathrooms %in% c(1.5, 2.0) ~ 2,
    bathrooms %in% c(2.5, 3.0) ~ 3,
    TRUE ~ 4
  ))
analysis %>%
  cbind(bathrooms_cut) %>%
  group_by(bathrooms_cut) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Guests_included

Distribuiton of Price by Guests Included

analysis %>%
  ggplot(aes(guests_included, price)) +
  geom_point() +
  geom_smooth(method = "lm", se = F)

## `geom_smooth()` using formula 'y ~ x'

Avg. Price with Guests_included

analysis %>%
  group_by(guests_included) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Feature Discre. - Guests_included

guests_cut <- analysis %>%
  summarize(guests_cut = cut(guests_included, seq(0, 18, 3)))
analysis %>%
  cbind(guests_cut) %>%
  group_by(guests_cut) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Extra_people

Distribution of Price by Extra People

analysis %>%
  ggplot(aes(extra_people, price)) +
  geom_point() +
  geom_smooth(method = "lm", se = F)

## `geom_smooth()` using formula 'y ~ x'

Avg. Price with Extra_people

analysis %>%
  group_by(extra_people) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Feature Discre. - Extra_people

extra_cut <- analysis %>%
  summarize(extra_cut = cut(extra_people, seq(0, 300, 50), include.lowest = T))
analysis %>%
  cbind(extra_cut) %>%
  group_by(extra_cut) %>%
  summarize(n = n(),
            mean_price = round(mean(price), 4)) %>%
  datatable()

Amenities

Since I believe amenities have an significant impact on price, I would like to first split them into separate words, then I would like to count the number of amenities each unit has to see whether it has a relationship with the price.

Separation of Amenities

library(wordcloud2)
# clean character
amenities_list <- str_split(analysis$amenities, ",")
amenities <- unlist(amenities_list)
amenities <- str_remove(amenities, fixed("."))
amenities <- amenities[amenities != ""]
amenities <- str_trim(amenities, "both")

# create word cloud
as.data.frame(amenities) %>%
  count(amenities, sort = T) %>%
  wordcloud2(size = 0.5)

# extract top10 amenities
amenities_top10 <- as.data.frame(amenities, stringsAsFactors = F) %>%
  count(amenities, sort = T) %>% head(10)
sort_amenities <- with(amenities_top10, reorder(amenities, n, mean))
ggplot(amenities_top10, aes(x = sort_amenities, weight = n)) +
  geom_bar(show.legend = F) +
  theme_bw() +
  coord_flip() +
  labs(title = "Top 10 Amenities",
       x = "",
       y = "")

Correlation Between Numeric Variables

Suddenly, I realize I could draw correlation plot between dependent variables rather than plotting one by one as there are variables with similar names, made me wondering if there are high correlation between them, so I would first subset similar variables, then check correlation one group after one group.

Subset Similar Variables

Availability Variables:

We can keep availability_30 and availability_365 and drop the others since most customers are looking for short-term rental.

Review Scores:

Everthing seems fine.

Host Listing Count:

We can see that calculated_host_lilstings_count is highly correlated with calculated_host_listings_count_entire_homes, we can drop the former.

Minimum & Maximum Nights:

We keep minimum_nights_avg_ntm and drop the others.

library(ggcorrplot)
# availability group
availability_group <- analysis %>% select(starts_with("availability"))
ggcorrplot(cor(availability_group), method = "square", type = "lower", show.diag = T, 
           colors = c("#6D9EC1", "white", "#E46726"), lab = T, digits = 3)

# review_scores group
review_scores_group <- analysis %>% select(starts_with("review_scores"))
ggcorrplot(cor(review_scores_group), method = "square", type = "lower", show.diag = T, 
           colors = c("#6D9EC1", "white", "#E46726"), lab = T, digits = 2)

# host_listing_count group
cal_host_listing_count_group <- analysis %>% select(starts_with("calculated"), host_total_listings_count)
ggcorrplot(cor(cal_host_listing_count_group), method = "square", type = "lower", show.diag = T, 
           colors = c("#6D9EC1", "white", "#E46726"), lab = T, digits = 2)

# minimum & maximum
minimum_maximum <- analysis %>% select(starts_with("minimum"), starts_with("maximum"))
ggcorrplot(cor(minimum_maximum), method = "square", type = "lower", show.diag = T, 
           colors = c("#6D9EC1", "white", "#E46726"), lab = T, digits = 2)

Correlation Between Other Numeric Variables

Now, we take a look at the correlation between other numeric variables. It seems there doesn’t have severe multicollinearity between these variables. However, cancellation_policy, reviews_per_month, number_of_reviews and extra_people have low correlation with price, we will examine these variables later in linear regression.

ggcorrplot(cor(analysis[, c("cleaning_fee", "accommodates", "bedrooms", "beds", "bathrooms", 
                            "guests_included", "extra_people", "number_of_reviews_ltm", 
                            "reviews_per_month", "first_review", "last_review", "host_since", 
                            "price")]), 
           method = "square", type = "lower",
           show.diag = T, colors = c("#6D9EC1", "white", "#E46726"), lab = T, digits = 1)

2nd Round of Data Cleaning

Analysis Data Cleaning

After EDA, we again found some unimportant variables and need to omit them from the analysis. So far, we have 37 variables including price in analysis. Now it’s time to build some models! Wait, we have to do the same thing to scoring_raw to ensure these two data files have the same structure.

# omit useless variables
analysis <- analysis %>%
  select(-host_has_profile_pic, -minimum_minimum_nights, -minimum_maximum_nights, -minimum_nights,
         -maximum_nights, -maximum_maximum_nights, -maximum_minimum_nights, 
         -maximum_nights_avg_ntm, -calculated_host_listings_count, 
         -availability_60, -availability_90)

# convert logical into factor
logic <- sapply(analysis, is.logical)
analysis[, logic] <- sapply(analysis[, logic], as.numeric)

# Convert Amenities into Number of Amenities
analysis$amenities <- sapply(amenities_list, length)

# double check
skim(analysis)

Data summary

Name	analysis
Number of rows	41322
Number of columns	37
_______________________
Column type frequency:
factor	4
numeric	33
________________________
Group variables	None

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
neighbourhood_group_cleansed	0	1	FALSE	5	Man: 17695, Bro: 17035, Que: 5194, Bro: 1056
property_type	0	1	FALSE	6	Apa: 32447, Hou: 5290, Con: 1393, Lof: 1226
room_type	0	1	FALSE	4	Ent: 21512, Pri: 18464, Sha: 1101, Hot: 245
bed_type	0	1	FALSE	5	Rea: 40759, Fut: 221, Pul: 189, Air: 105

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
price	0	1	137.04	111.06	10.00	66.00	100.00	170	999.00	▇▁▁▁▁
cleaning_fee	0	1	58.17	52.33	1.58	20.00	50.00	80	600.00	▇▁▁▁▁
beds	0	1	1.58	1.15	0.00	1.00	1.00	2	26.00	▇▁▁▁▁
host_since	0	1	1987.05	915.91	209.00	1267.00	2006.00	2683	4509.00	▅▆▇▅▁
host_is_superhost	0	1	0.22	0.41	0.00	0.00	0.00	0	1.00	▇▁▁▁▂
host_total_listings_count	0	1	8.79	72.53	0.00	1.00	1.00	2	2345.00	▇▁▁▁▁
host_identity_verified	0	1	0.46	0.50	0.00	0.00	0.00	1	1.00	▇▁▁▁▇
first_review	0	1	1185.33	712.32	206.00	594.00	999.00	1640	4329.00	▇▅▂▁▁
last_review	0	1	628.51	437.25	205.00	333.00	455.00	729	3521.00	▇▁▁▁▁
reviews_per_month	0	1	1.35	1.59	0.01	0.22	0.76	2	33.71	▇▁▁▁▁
is_location_exact	0	1	0.83	0.38	0.00	1.00	1.00	1	1.00	▂▁▁▁▇
accommodates	0	1	2.93	1.91	1.00	2.00	2.00	4	18.00	▇▁▁▁▁
bathrooms	0	1	1.14	0.41	0.00	1.00	1.00	1	7.00	▇▁▁▁▁
bedrooms	0	1	1.19	0.75	0.00	1.00	1.00	1	21.00	▇▁▁▁▁
amenities	0	1	22.42	9.27	1.00	15.00	21.00	29	83.00	▅▇▂▁▁
guests_included	0	1	1.60	1.23	1.00	1.00	1.00	2	16.00	▇▁▁▁▁
extra_people	0	1	16.19	24.94	0.00	0.00	10.00	25	300.00	▇▁▁▁▁
minimum_nights_avg_ntm	0	1	6.96	29.80	1.00	1.30	2.00	4	1587.10	▇▁▁▁▁
availability_30	0	1	11.20	11.76	0.00	0.00	7.00	23	30.00	▇▂▂▂▃
availability_365	0	1	126.16	132.62	0.00	0.00	79.00	245	365.00	▇▂▂▁▃
number_of_reviews_ltm	0	1	11.21	16.78	0.00	1.00	4.00	16	351.00	▇▁▁▁▁
review_scores_rating	0	1	93.61	9.37	20.00	91.00	96.00	100	100.00	▁▁▁▁▇
review_scores_accuracy	0	1	9.58	0.92	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
review_scores_cleanliness	0	1	9.25	1.13	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
review_scores_checkin	0	1	9.71	0.80	2.00	10.00	10.00	10	10.00	▁▁▁▁▇
review_scores_communication	0	1	9.72	0.82	2.00	10.00	10.00	10	10.00	▁▁▁▁▇
review_scores_location	0	1	9.58	0.78	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
review_scores_value	0	1	9.37	0.97	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
instant_bookable	0	1	0.40	0.49	0.00	0.00	0.00	1	1.00	▇▁▁▁▅
cancellation_policy	0	1	1.71	0.82	1.00	1.00	1.00	2	3.00	▇▁▃▁▃
calculated_host_listings_count_entire_homes	0	1	4.03	26.88	0.00	0.00	1.00	1	319.00	▇▁▁▁▁
calculated_host_listings_count_private_rooms	0	1	1.44	6.27	0.00	0.00	0.00	1	150.00	▇▁▁▁▁
calculated_host_listings_count_shared_rooms	0	1	0.14	1.04	0.00	0.00	0.00	0	24.00	▇▁▁▁▁

Scoring Data Cleaning

Read Data

scoring_raw <- read.csv("scoringData.csv")

Routine Data Cleaning

First, I would create a price column for convenience as I would use the names of variables from analysis to select necessary variables.

Then, let’s take a look at the data. We can see that some there are 10 character variables, and we need to convert them into the same type as from analysis.

Beautiful!

scoring <- scoring_raw %>%
  # create a price column for convenience
  mutate(price = 1) %>%
  # select chosen variables from analysis
  select(id, names(analysis)) %>%
  # change types of variables
  mutate(
    host_is_superhost = as.numeric(ifelse(host_is_superhost == "f", 0, 1)),
    host_identity_verified = as.numeric(ifelse(host_identity_verified == "f", 0, 1)),
    is_location_exact = as.numeric(ifelse(is_location_exact == "f", 0, 1)),
    instant_bookable = as.numeric(ifelse(instant_bookable == "f", 0, 1)),
    # host_has_profile_pic = as.numeric(ifelse(host_has_profile_pic == "f", 0, 1)),
    bed_type = as.factor(bed_type)
  ) %>%
  # change character to factor
  mutate(
    neighbourhood_group_cleansed = as.factor(neighbourhood_group_cleansed),
    property_type = as.factor(property_type),
    room_type = as.factor(room_type),
    cancellation_policy = as.factor(cancellation_policy)
  ) %>%
  # collapse property_type factors
  mutate(
    property_type = fct_collapse(property_type, 
                                 "Apartment" = c("Apartment", "Serviced apartment"),
                                 "Condominium" = "Condominium",
                                 "House" = c("Bungalow", "House", "Guesthouse", "Hostel", 
                                             "Tiny house", "Townhouse", "Lighthouse", 
                                             "Dome house", "Earth house", "Villa"),
                                 "Loft" = "Loft",
                                 "Hotel" = c("Hotel", "Bed and breakfast", "Boutique hotel", 
                                             "Guest suite", "Resort"))
    ) %>%
  mutate(property_type = fct_other(property_type, keep = c("Apartment", "Condominium", "House", 
                                                           "Loft", "Hotel"))) %>%
  mutate(cancellation_policy = fct_collapse(cancellation_policy, 
                                       "strict" = c("strict", "super_strict_30", "super_strict_60",
                                                    "strict_14_with_grace_period"))) %>%
  # transform cancellation_policy into dummy variables
  mutate(cancellation_policy = case_when(
    cancellation_policy == "flexible" ~ 3,
    cancellation_policy == "moderate" ~ 2,
    cancellation_policy == "strict" ~ 1
   )) %>%
  # fill missing values in host_total_listings_count
  fill(host_total_listings_count) %>%
  # remove column price
  select(-price) %>%
  # convert data into period
  mutate(last_review = as.Date(last_review),
         host_since = as.Date(host_since)) %>%
  as_tibble()

# change first_review
scoring$first_review = as.numeric(floor(difftime("2020-12-31", scoring$first_review, units = "days")))
# change last_review
scoring$last_review = as.numeric(floor(difftime("2020-12-31", scoring$last_review, units = "days")))
# change host_since
scoring$host_since = as.numeric(floor(difftime("2020-12-31", scoring$host_since, units = "days")))

# fill missing values in beds with ceiling(mean(beds))
scoring$beds[is.na(scoring$beds)] = ceiling(mean(scoring$beds, na.rm = T))
# fill missing values in cleaning_fee with avg. cleaning_fee
scoring$cleaning_fee[is.na(scoring$cleaning_fee)] = mean(scoring$cleaning_fee, na.rm = T)
# fill missing values in host_since
scoring$host_since[is.na(scoring$host_since)] = floor(mean(scoring$host_since, na.rm = T))

# convert integer into numeric
int <- unlist(lapply(scoring, is.integer))
scoring[, int] <- lapply(scoring[, int], as.numeric)

# double check
datatable(miss_var_summary(scoring))

Convert Amenities into Number of Amenities

For now, scoring data has the same data structure as analysis data set. Good job!

amenities_sc_list <- str_split(scoring$amenities, ", ")
scoring$amenities <- sapply(amenities_sc_list, length)
skim(scoring)

Data summary

Name	scoring
Number of rows	10333
Number of columns	37
_______________________
Column type frequency:
factor	4
numeric	33
________________________
Group variables	None

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
neighbourhood_group_cleansed	0	1	FALSE	5	Man: 4442, Bro: 4194, Que: 1382, Bro: 233
property_type	0	1	FALSE	6	Apa: 8104, Hou: 1301, Con: 362, Lof: 309
room_type	0	1	FALSE	4	Ent: 5320, Pri: 4722, Sha: 225, Hot: 66
bed_type	0	1	FALSE	5	Rea: 10185, Fut: 57, Pul: 49, Air: 34

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
id	0	1	453411.30	271605.54	10310.00	226856.00	431905.00	670735	998507.00	▇▇▇▅▅
cleaning_fee	0	1	64.09	49.36	0.00	30.00	60.00	80	600.00	▇▁▁▁▁
beds	0	1	1.58	1.10	0.00	1.00	1.00	2	16.00	▇▁▁▁▁
host_since	0	1	1973.37	905.96	209.00	1259.00	1986.00	2646	4496.00	▅▆▇▅▁
host_is_superhost	0	1	0.22	0.41	0.00	0.00	0.00	0	1.00	▇▁▁▁▂
host_total_listings_count	0	1	9.67	79.60	0.00	1.00	1.00	2	2345.00	▇▁▁▁▁
host_identity_verified	0	1	0.46	0.50	0.00	0.00	0.00	1	1.00	▇▁▁▁▇
first_review	0	1	1181.06	704.01	206.00	599.00	992.00	1634	4106.00	▇▅▂▁▁
last_review	0	1	626.32	439.37	206.00	332.00	457.00	726	3391.00	▇▁▁▁▁
reviews_per_month	0	1	1.36	1.71	0.01	0.22	0.76	2	66.36	▇▁▁▁▁
is_location_exact	0	1	0.83	0.37	0.00	1.00	1.00	1	1.00	▂▁▁▁▇
accommodates	0	1	2.92	1.90	1.00	2.00	2.00	4	16.00	▇▁▁▁▁
bathrooms	0	1	1.15	0.43	0.00	1.00	1.00	1	7.50	▇▁▁▁▁
bedrooms	0	1	1.19	0.74	0.00	1.00	1.00	1	9.00	▇▂▁▁▁
amenities	0	1	12.96	6.47	1.00	8.00	12.00	17	50.00	▇▇▂▁▁
guests_included	0	1	1.58	1.18	1.00	1.00	1.00	2	16.00	▇▁▁▁▁
extra_people	0	1	15.82	24.49	0.00	0.00	10.00	25	300.00	▇▁▁▁▁
minimum_nights_avg_ntm	0	1	7.40	34.61	1.00	1.10	2.00	4	1124.00	▇▁▁▁▁
availability_30	0	1	11.20	11.71	0.00	0.00	7.00	23	30.00	▇▂▂▂▃
availability_365	0	1	127.30	132.87	0.00	0.00	80.00	249	365.00	▇▂▂▁▃
number_of_reviews_ltm	0	1	11.46	18.48	0.00	1.00	4.00	15	730.00	▇▁▁▁▁
review_scores_rating	0	1	93.52	9.44	20.00	91.00	96.00	100	100.00	▁▁▁▁▇
review_scores_accuracy	0	1	9.57	0.93	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
review_scores_cleanliness	0	1	9.24	1.14	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
review_scores_checkin	0	1	9.71	0.79	2.00	10.00	10.00	10	10.00	▁▁▁▁▇
review_scores_communication	0	1	9.71	0.83	2.00	10.00	10.00	10	10.00	▁▁▁▁▇
review_scores_location	0	1	9.58	0.80	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
review_scores_value	0	1	9.34	1.00	2.00	9.00	10.00	10	10.00	▁▁▁▁▇
instant_bookable	0	1	0.39	0.49	0.00	0.00	0.00	1	1.00	▇▁▁▁▅
cancellation_policy	0	1	1.71	0.82	1.00	1.00	1.00	2	3.00	▇▁▃▁▃
calculated_host_listings_count_entire_homes	0	1	4.25	27.58	0.00	0.00	1.00	1	319.00	▇▁▁▁▁
calculated_host_listings_count_private_rooms	0	1	1.47	6.22	0.00	0.00	1.00	1	133.00	▇▁▁▁▁
calculated_host_listings_count_shared_rooms	0	1	0.12	0.99	0.00	0.00	0.00	0	24.00	▇▁▁▁▁

Split Data

First, split the data into 70% to the train set, 30% to the test set.

set.seed(1031)
split = createDataPartition(y = analysis$price, p = 0.7, list = F, groups = 10)
train = analysis[split, ]
test = analysis[-split, ]

Linear Regression

First, let’s use the simple linear regression to see the performance. The results are shown below:
R2: 0.5309
RSE: 75.38
RMSE_test: 78.33

mod1 = lm(price ~ ., data = train)
summary(mod1)

## 
## Call:
## lm(formula = price ~ ., data = train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -456.29  -37.07   -6.48   23.52  836.92 
## 
## Coefficients:
##                                                Estimate Std. Error t value
## (Intercept)                                  -1.282e+02  1.174e+01 -10.920
## cleaning_fee                                  3.112e-01  1.156e-02  26.921
## beds                                         -3.107e+00  6.274e-01  -4.952
## host_since                                   -1.207e-04  6.627e-04  -0.182
## host_is_superhost                             4.790e+00  1.219e+00   3.928
## host_total_listings_count                     3.689e-02  8.286e-03   4.452
## host_identity_verified                        4.332e-01  1.036e+00   0.418
## first_review                                 -1.985e-03  8.919e-04  -2.226
## last_review                                   1.807e-02  1.373e-03  13.156
## reviews_per_month                            -1.938e-01  5.252e-01  -0.369
## neighbourhood_group_cleansedBrooklyn          1.981e+01  2.899e+00   6.834
## neighbourhood_group_cleansedManhattan         6.696e+01  2.940e+00  22.777
## neighbourhood_group_cleansedQueens            9.243e+00  3.050e+00   3.030
## neighbourhood_group_cleansedStaten Island    -1.964e+01  5.629e+00  -3.488
## is_location_exact                            -4.194e+00  1.183e+00  -3.547
## property_typeHotel                            3.125e+01  3.587e+00   8.711
## property_typeHouse                           -2.197e+00  1.493e+00  -1.471
## property_typeCondominium                      2.524e+01  2.482e+00  10.168
## property_typeLoft                             4.194e+01  2.683e+00  15.631
## property_typeOther                            2.617e+01  7.774e+00   3.367
## room_typeHotel room                           1.684e+00  6.582e+00   0.256
## room_typePrivate room                        -5.123e+01  1.180e+00 -43.427
## room_typeShared room                         -6.841e+01  3.718e+00 -18.398
## accommodates                                  1.299e+01  4.425e-01  29.366
## bathrooms                                     3.168e+01  1.218e+00  26.019
## bedrooms                                      2.181e+01  8.762e-01  24.888
## bed_typeCouch                                 1.759e+01  1.652e+01   1.065
## bed_typeFuton                                 5.620e+00  1.063e+01   0.529
## bed_typePull-out Sofa                         1.621e+00  1.109e+01   0.146
## bed_typeReal Bed                              5.814e+00  8.747e+00   0.665
## amenities                                     1.789e-01  5.735e-02   3.120
## guests_included                               2.841e+00  4.830e-01   5.881
## extra_people                                  3.466e-02  1.941e-02   1.786
## minimum_nights_avg_ntm                       -1.964e-01  1.752e-02 -11.210
## availability_30                               4.563e-01  4.949e-02   9.221
## availability_365                              3.249e-02  4.481e-03   7.252
## number_of_reviews_ltm                        -3.739e-01  4.992e-02  -7.491
## review_scores_rating                          8.626e-01  1.029e-01   8.383
## review_scores_accuracy                       -3.558e+00  8.335e-01  -4.269
## review_scores_cleanliness                     3.958e+00  6.022e-01   6.573
## review_scores_checkin                        -2.724e+00  8.186e-01  -3.327
## review_scores_communication                  -1.225e+00  8.618e-01  -1.421
## review_scores_location                        1.269e+01  6.974e-01  18.194
## review_scores_value                          -6.128e+00  7.924e-01  -7.733
## instant_bookable                              4.781e+00  9.831e-01   4.863
## cancellation_policy                           1.393e-01  5.852e-01   0.238
## calculated_host_listings_count_entire_homes  -1.441e-02  2.145e-02  -0.672
## calculated_host_listings_count_private_rooms -2.173e-01  7.300e-02  -2.976
## calculated_host_listings_count_shared_rooms  -1.909e+00  5.503e-01  -3.468
##                                              Pr(>|t|)    
## (Intercept)                                   < 2e-16 ***
## cleaning_fee                                  < 2e-16 ***
## beds                                         7.39e-07 ***
## host_since                                   0.855436    
## host_is_superhost                            8.60e-05 ***
## host_total_listings_count                    8.53e-06 ***
## host_identity_verified                       0.675901    
## first_review                                 0.026047 *  
## last_review                                   < 2e-16 ***
## reviews_per_month                            0.712064    
## neighbourhood_group_cleansedBrooklyn         8.40e-12 ***
## neighbourhood_group_cleansedManhattan         < 2e-16 ***
## neighbourhood_group_cleansedQueens           0.002445 ** 
## neighbourhood_group_cleansedStaten Island    0.000487 ***
## is_location_exact                            0.000391 ***
## property_typeHotel                            < 2e-16 ***
## property_typeHouse                           0.141221    
## property_typeCondominium                      < 2e-16 ***
## property_typeLoft                             < 2e-16 ***
## property_typeOther                           0.000762 ***
## room_typeHotel room                          0.798052    
## room_typePrivate room                         < 2e-16 ***
## room_typeShared room                          < 2e-16 ***
## accommodates                                  < 2e-16 ***
## bathrooms                                     < 2e-16 ***
## bedrooms                                      < 2e-16 ***
## bed_typeCouch                                0.287079    
## bed_typeFuton                                0.597121    
## bed_typePull-out Sofa                        0.883787    
## bed_typeReal Bed                             0.506223    
## amenities                                    0.001810 ** 
## guests_included                              4.11e-09 ***
## extra_people                                 0.074087 .  
## minimum_nights_avg_ntm                        < 2e-16 ***
## availability_30                               < 2e-16 ***
## availability_365                             4.22e-13 ***
## number_of_reviews_ltm                        7.02e-14 ***
## review_scores_rating                          < 2e-16 ***
## review_scores_accuracy                       1.97e-05 ***
## review_scores_cleanliness                    5.00e-11 ***
## review_scores_checkin                        0.000878 ***
## review_scores_communication                  0.155229    
## review_scores_location                        < 2e-16 ***
## review_scores_value                          1.08e-14 ***
## instant_bookable                             1.16e-06 ***
## cancellation_policy                          0.811792    
## calculated_host_listings_count_entire_homes  0.501551    
## calculated_host_listings_count_private_rooms 0.002919 ** 
## calculated_host_listings_count_shared_rooms  0.000525 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 75.38 on 28881 degrees of freedom
## Multiple R-squared:  0.5309, Adjusted R-squared:  0.5301 
## F-statistic: 680.8 on 48 and 28881 DF,  p-value: < 2.2e-16

# predict using test
pred_mod1_test = predict(mod1, newdata = test)
rmse_mod1_test = sqrt(mean((pred_mod1_test - test$price)^2)); rmse_mod1_test

## [1] 78.32862

Forward Selection

First, let’s use forward selection. We can see that it didn’t select reviews_per_month, bed_type, cancellation_policy, calculated_host_listings_count_entire_homes, and host_since.

R2: 0.5308
RSE: 75.37
RMSE_test: 78.31

start_mod = lm(price ~ 1, train)
empty_mod = lm(price ~ 1, train)
full_mod = lm(price ~ ., train)
forward_selection = step(start_mod,
                      scope = list(upper = full_mod, lower = empty_mod),
                      direction ='forward')

## Start:  AIC=271949.8
## price ~ 1
## 
##                                                Df Sum of Sq       RSS    AIC
## + accommodates                                  1 107571084 242215318 261320
## + cleaning_fee                                  1  92656648 257129754 263049
## + room_type                                     3  88187304 261599099 263551
## + bedrooms                                      1  64299037 285487366 266075
## + beds                                          1  60871918 288914485 266421
## + guests_included                               1  48282114 301504288 267655
## + neighbourhood_group_cleansed                  4  31684570 318101832 269211
## + bathrooms                                     1  24146915 325639488 269882
## + amenities                                     1  10778512 339007890 271046
## + property_type                                 5   7760124 342026279 271311
## + review_scores_location                        1   7039169 342747234 271364
## + cancellation_policy                           1   6265786 343520617 271429
## + calculated_host_listings_count_private_rooms  1   4657013 345129390 271564
## + calculated_host_listings_count_entire_homes   1   3844913 345941490 271632
## + availability_365                              1   3532874 346253528 271658
## + extra_people                                  1   3465224 346321179 271664
## + review_scores_cleanliness                     1   3248152 346538251 271682
## + calculated_host_listings_count_shared_rooms   1   3089088 346697314 271695
## + review_scores_rating                          1   1921937 347864465 271792
## + host_total_listings_count                     1   1849860 347936542 271798
## + availability_30                               1   1141321 348645081 271857
## + bed_type                                      4    473718 349312684 271919
## + number_of_reviews_ltm                         1    380944 349405459 271920
## + review_scores_accuracy                        1    365262 349421141 271922
## + review_scores_communication                   1    311156 349475247 271926
## + minimum_nights_avg_ntm                        1    169924 349616479 271938
## + review_scores_checkin                         1    159270 349627132 271939
## + host_is_superhost                             1    150457 349635945 271939
## + last_review                                   1     77984 349708419 271945
## + reviews_per_month                             1     77029 349709373 271945
## + is_location_exact                             1     37944 349748459 271949
## + host_since                                    1     31445 349754957 271949
## <none>                                                      349786403 271950
## + host_identity_verified                        1      6235 349780168 271951
## + instant_bookable                              1      3240 349783162 271951
## + review_scores_value                           1      2704 349783699 271952
## + first_review                                  1         2 349786401 271952
## 
## Step:  AIC=261320.1
## price ~ accommodates
## 
##                                                Df Sum of Sq       RSS    AIC
## + neighbourhood_group_cleansed                  4  32174958 210040360 257205
## + cleaning_fee                                  1  28544773 213670545 257695
## + room_type                                     3  25374727 216840592 258125
## + property_type                                 5   8865091 233350227 260251
## + review_scores_location                        1   7317323 234897995 260435
## + reviews_per_month                             1   2890939 239324379 260975
## + number_of_reviews_ltm                         1   2886403 239328916 260975
## + bathrooms                                     1   2665346 239549972 261002
## + bedrooms                                      1   2163182 240052137 261063
## + review_scores_rating                          1   2129729 240085589 261067
## + review_scores_cleanliness                     1   2116216 240099103 261068
## + calculated_host_listings_count_entire_homes   1   2106238 240109080 261069
## + host_total_listings_count                     1   1793831 240421487 261107
## + cancellation_policy                           1   1309853 240905466 261165
## + calculated_host_listings_count_private_rooms  1   1221844 240993474 261176
## + calculated_host_listings_count_shared_rooms   1    958515 241256803 261207
## + host_since                                    1    849716 241365603 261220
## + last_review                                   1    791425 241423893 261227
## + guests_included                               1    745200 241470118 261233
## + extra_people                                  1    704055 241511263 261238
## + review_scores_accuracy                        1    587211 241628107 261252
## + amenities                                     1    562447 241652871 261255
## + review_scores_communication                   1    549689 241665629 261256
## + instant_bookable                              1    324896 241890422 261283
## + minimum_nights_avg_ntm                        1    318370 241896948 261284
## + host_identity_verified                        1    267355 241947964 261290
## + first_review                                  1    266164 241949154 261290
## + review_scores_checkin                         1    172833 242042486 261301
## + review_scores_value                           1    148341 242066978 261304
## + availability_365                              1    141164 242074154 261305
## <none>                                                      242215318 261320
## + availability_30                               1     12339 242202979 261321
## + beds                                          1     10117 242205201 261321
## + host_is_superhost                             1      4345 242210974 261322
## + is_location_exact                             1       105 242215213 261322
## + bed_type                                      4     47782 242167536 261322
## 
## Step:  AIC=257204.8
## price ~ accommodates + neighbourhood_group_cleansed
## 
##                                                Df Sum of Sq       RSS    AIC
## + cleaning_fee                                  1  17409629 192630732 254704
## + room_type                                     3  16181402 193858958 254892
## + property_type                                 5   4226454 205813906 256627
## + bedrooms                                      1   4083693 205956667 256639
## + review_scores_location                        1   3902797 206137563 256664
## + bathrooms                                     1   3647431 206392930 256700
## + review_scores_cleanliness                     1   2628275 207412086 256842
## + review_scores_rating                          1   2420269 207620091 256871
## + reviews_per_month                             1   1635941 208404419 256981
## + number_of_reviews_ltm                         1   1632052 208408308 256981
## + guests_included                               1   1328023 208712337 257023
## + amenities                                     1    904330 209136030 257082
## + review_scores_accuracy                        1    795329 209245031 257097
## + calculated_host_listings_count_shared_rooms   1    771263 209269097 257100
## + host_total_listings_count                     1    702403 209337957 257110
## + review_scores_communication                   1    630659 209409701 257120
## + calculated_host_listings_count_private_rooms  1    548508 209491852 257131
## + availability_365                              1    535010 209505350 257133
## + review_scores_value                           1    493131 209547229 257139
## + calculated_host_listings_count_entire_homes   1    491027 209549334 257139
## + extra_people                                  1    455926 209584435 257144
## + host_since                                    1    431149 209609211 257147
## + review_scores_checkin                         1    400701 209639659 257152
## + cancellation_policy                           1    348477 209691883 257159
## + instant_bookable                              1    244602 209795758 257173
## + availability_30                               1    205890 209834470 257178
## + last_review                                   1    191292 209849069 257180
## + host_identity_verified                        1    127537 209912823 257189
## + host_is_superhost                             1    117438 209922922 257191
## + beds                                          1    103574 209936786 257193
## + first_review                                  1     48314 209992046 257200
## + minimum_nights_avg_ntm                        1     34102 210006259 257202
## <none>                                                      210040360 257205
## + bed_type                                      4     54998 209985362 257205
## + is_location_exact                             1       291 210040069 257207
## 
## Step:  AIC=254703.6
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee
## 
##                                                Df Sum of Sq       RSS    AIC
## + room_type                                     3   9076652 183554079 253313
## + property_type                                 5   3280980 189349752 254217
## + review_scores_location                        1   2798479 189832252 254282
## + bathrooms                                     1   2676811 189953920 254301
## + bedrooms                                      1   2576486 190054246 254316
## + review_scores_rating                          1   1647766 190982966 254457
## + review_scores_cleanliness                     1   1526811 191103920 254475
## + number_of_reviews_ltm                         1   1167992 191462740 254530
## + last_review                                   1    907829 191722903 254569
## + reviews_per_month                             1    878743 191751988 254573
## + review_scores_accuracy                        1    536474 192094257 254625
## + review_scores_communication                   1    535801 192094931 254625
## + review_scores_value                           1    500955 192129776 254630
## + calculated_host_listings_count_shared_rooms   1    420772 192209960 254642
## + minimum_nights_avg_ntm                        1    416241 192214490 254643
## + guests_included                               1    409324 192221407 254644
## + calculated_host_listings_count_private_rooms  1    334953 192295779 254655
## + review_scores_checkin                         1    301046 192329686 254660
## + host_since                                    1     95911 192534821 254691
## + cancellation_policy                           1     77987 192552745 254694
## + host_identity_verified                        1     67267 192563464 254696
## + instant_bookable                              1     57738 192572994 254697
## + availability_30                               1     54902 192575829 254697
## + availability_365                              1     41068 192589664 254699
## + first_review                                  1     24778 192605954 254702
## + calculated_host_listings_count_entire_homes   1     22938 192607793 254702
## + amenities                                     1     19316 192611415 254703
## + extra_people                                  1     15893 192614839 254703
## <none>                                                      192630732 254704
## + is_location_exact                             1     11940 192618791 254704
## + host_is_superhost                             1      1667 192629064 254705
## + beds                                          1      1652 192629079 254705
## + host_total_listings_count                     1       709 192630022 254706
## + bed_type                                      4     34284 192596447 254706
## 
## Step:  AIC=253313.3
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type
## 
##                                                Df Sum of Sq       RSS    AIC
## + bathrooms                                     1   5445978 178108101 252444
## + bedrooms                                      1   5010716 178543363 252515
## + property_type                                 5   2611161 180942918 252909
## + review_scores_location                        1   2093044 181461035 252984
## + review_scores_rating                          1   1161278 182392801 253132
## + review_scores_cleanliness                     1   1056034 182498045 253148
## + number_of_reviews_ltm                         1   1051395 182502684 253149
## + reviews_per_month                             1    637271 182916808 253215
## + last_review                                   1    583557 182970522 253223
## + minimum_nights_avg_ntm                        1    480075 183074004 253240
## + availability_30                               1    442656 183111423 253245
## + guests_included                               1    385201 183168878 253255
## + availability_365                              1    347402 183206677 253260
## + review_scores_value                           1    338100 183215980 253262
## + review_scores_accuracy                        1    258519 183295560 253275
## + review_scores_communication                   1    256298 183297781 253275
## + review_scores_checkin                         1    155673 183398406 253291
## + beds                                          1    141614 183412465 253293
## + extra_people                                  1    111109 183442970 253298
## + is_location_exact                             1     32804 183521275 253310
## + cancellation_policy                           1     28258 183525822 253311
## + calculated_host_listings_count_shared_rooms   1     14008 183540071 253313
## + calculated_host_listings_count_private_rooms  1     12983 183541096 253313
## <none>                                                      183554079 253313
## + amenities                                     1     12037 183542042 253313
## + first_review                                  1      3875 183550204 253315
## + host_identity_verified                        1      2530 183551549 253315
## + calculated_host_listings_count_entire_homes   1      1114 183552965 253315
## + host_is_superhost                             1      1052 183553027 253315
## + host_since                                    1       279 183553801 253315
## + instant_bookable                              1       119 183553960 253315
## + host_total_listings_count                     1        84 183553995 253315
## + bed_type                                      4     21800 183532280 253318
## 
## Step:  AIC=252444
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms
## 
##                                                Df Sum of Sq       RSS    AIC
## + bedrooms                                      1   3346601 174761501 251897
## + property_type                                 5   2598934 175509167 252029
## + review_scores_location                        1   2229459 175878642 252082
## + review_scores_cleanliness                     1   1245236 176862865 252243
## + review_scores_rating                          1   1231498 176876604 252245
## + number_of_reviews_ltm                         1    841037 177267065 252309
## + minimum_nights_avg_ntm                        1    517300 177590801 252362
## + availability_30                               1    505092 177603009 252364
## + last_review                                   1    502429 177605673 252364
## + reviews_per_month                             1    470097 177638005 252370
## + review_scores_value                           1    353682 177754419 252388
## + review_scores_communication                   1    351727 177756374 252389
## + guests_included                               1    317022 177791080 252394
## + availability_365                              1    311219 177796883 252395
## + review_scores_accuracy                        1    299324 177808778 252397
## + review_scores_checkin                         1    186421 177921680 252416
## + extra_people                                  1    136988 177971113 252424
## + calculated_host_listings_count_shared_rooms   1     83785 178024317 252432
## + calculated_host_listings_count_private_rooms  1     80368 178027733 252433
## + is_location_exact                             1     53768 178054334 252437
## + cancellation_policy                           1     32078 178076023 252441
## + host_identity_verified                        1     13211 178094890 252444
## <none>                                                      178108101 252444
## + host_is_superhost                             1     12125 178095976 252444
## + amenities                                     1      7379 178100723 252445
## + beds                                          1      5749 178102352 252445
## + host_since                                    1      3324 178104777 252445
## + instant_bookable                              1       799 178107303 252446
## + calculated_host_listings_count_entire_homes   1       587 178107514 252446
## + first_review                                  1       191 178107910 252446
## + host_total_listings_count                     1        59 178108043 252446
## + bed_type                                      4     18205 178089897 252449
## 
## Step:  AIC=251897.2
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms
## 
##                                                Df Sum of Sq       RSS    AIC
## + property_type                                 5   2881333 171880167 251426
## + review_scores_location                        1   2216952 172544549 251530
## + review_scores_cleanliness                     1   1210719 173550782 251698
## + review_scores_rating                          1   1129902 173631599 251712
## + number_of_reviews_ltm                         1    718391 174043110 251780
## + availability_30                               1    598770 174162731 251800
## + minimum_nights_avg_ntm                        1    472047 174289454 251821
## + availability_365                              1    425080 174336421 251829
## + last_review                                   1    406854 174354646 251832
## + reviews_per_month                             1    378593 174382907 251836
## + review_scores_communication                   1    290042 174471459 251851
## + review_scores_value                           1    286880 174474621 251852
## + review_scores_accuracy                        1    262132 174499369 251856
## + beds                                          1    195030 174566471 251867
## + review_scores_checkin                         1    150130 174611371 251874
## + guests_included                               1    132396 174629105 251877
## + extra_people                                  1    120327 174641173 251879
## + calculated_host_listings_count_shared_rooms   1     82805 174678696 251886
## + calculated_host_listings_count_private_rooms  1     72162 174689339 251887
## + is_location_exact                             1     61507 174699993 251889
## + calculated_host_listings_count_entire_homes   1     19995 174741506 251896
## + amenities                                     1     18990 174742511 251896
## + host_is_superhost                             1     16225 174745276 251897
## + host_total_listings_count                     1     15052 174746448 251897
## + cancellation_policy                           1     12755 174748746 251897
## <none>                                                      174761501 251897
## + instant_bookable                              1     11060 174750440 251897
## + host_identity_verified                        1      6914 174754586 251898
## + first_review                                  1      5063 174756438 251898
## + host_since                                    1         5 174761495 251899
## + bed_type                                      4     11787 174749714 251903
## 
## Step:  AIC=251426.3
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type
## 
##                                                Df Sum of Sq       RSS    AIC
## + review_scores_location                        1   2036782 169843385 251083
## + review_scores_cleanliness                     1   1141212 170738956 251236
## + review_scores_rating                          1   1068460 170811707 251248
## + number_of_reviews_ltm                         1    703146 171177022 251310
## + availability_30                               1    537325 171342843 251338
## + last_review                                   1    493733 171386435 251345
## + minimum_nights_avg_ntm                        1    435710 171444457 251355
## + availability_365                              1    402712 171477455 251360
## + reviews_per_month                             1    368878 171511289 251366
## + review_scores_communication                   1    292782 171587386 251379
## + review_scores_value                           1    273216 171606952 251382
## + review_scores_accuracy                        1    265450 171614717 251384
## + review_scores_checkin                         1    157566 171722601 251402
## + beds                                          1    150923 171729245 251403
## + guests_included                               1    134569 171745598 251406
## + extra_people                                  1     86235 171793933 251414
## + calculated_host_listings_count_shared_rooms   1     77472 171802696 251415
## + is_location_exact                             1     69320 171810847 251417
## + calculated_host_listings_count_private_rooms  1     68944 171811224 251417
## + calculated_host_listings_count_entire_homes   1     37496 171842672 251422
## + host_total_listings_count                     1     24751 171855416 251424
## + host_is_superhost                             1     14164 171866003 251426
## <none>                                                      171880167 251426
## + instant_bookable                              1     10118 171870049 251427
## + cancellation_policy                           1      9068 171871099 251427
## + host_identity_verified                        1      7248 171872919 251427
## + amenities                                     1       877 171879290 251428
## + first_review                                  1       789 171879379 251428
## + host_since                                    1       787 171879381 251428
## + bed_type                                      4      9946 171870222 251433
## 
## Step:  AIC=251083.4
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location
## 
##                                                Df Sum of Sq       RSS    AIC
## + number_of_reviews_ltm                         1    895490 168947895 250932
## + last_review                                   1    715884 169127501 250963
## + availability_30                               1    597651 169245734 250983
## + availability_365                              1    540236 169303150 250993
## + reviews_per_month                             1    531091 169312295 250995
## + minimum_nights_avg_ntm                        1    381715 169461670 251020
## + review_scores_cleanliness                     1    317543 169525842 251031
## + guests_included                               1    145137 169698248 251061
## + review_scores_rating                          1    140101 169703284 251062
## + beds                                          1    131157 169712228 251063
## + extra_people                                  1     76874 169766512 251072
## + is_location_exact                             1     76440 169766945 251072
## + review_scores_checkin                         1     61200 169782185 251075
## + calculated_host_listings_count_shared_rooms   1     61188 169782197 251075
## + review_scores_value                           1     56939 169786446 251076
## + calculated_host_listings_count_private_rooms  1     47145 169796240 251077
## + calculated_host_listings_count_entire_homes   1     38886 169804499 251079
## + host_total_listings_count                     1     36234 169807151 251079
## + review_scores_accuracy                        1     30615 169812770 251080
## + instant_bookable                              1     14246 169829139 251083
## + review_scores_communication                   1     13686 169829699 251083
## <none>                                                      169843385 251083
## + cancellation_policy                           1      5686 169837699 251084
## + host_since                                    1      4743 169838642 251085
## + host_identity_verified                        1      4325 169839061 251085
## + amenities                                     1      4039 169839346 251085
## + first_review                                  1      2493 169840893 251085
## + host_is_superhost                             1      1125 169842261 251085
## + bed_type                                      4      9044 169834341 251090
## 
## Step:  AIC=250932.5
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm
## 
##                                                Df Sum of Sq       RSS    AIC
## + availability_30                               1   1024666 167923229 250758
## + availability_365                              1    797741 168150154 250798
## + minimum_nights_avg_ntm                        1    469247 168478649 250854
## + review_scores_cleanliness                     1    399665 168548230 250866
## + last_review                                   1    286182 168661714 250885
## + guests_included                               1    207117 168740779 250899
## + review_scores_rating                          1    156309 168791586 250908
## + extra_people                                  1    118321 168829575 250914
## + beds                                          1    112281 168835614 250915
## + host_is_superhost                             1    107063 168840832 250916
## + is_location_exact                             1     87640 168860255 250919
## + calculated_host_listings_count_shared_rooms   1     57660 168890236 250925
## + calculated_host_listings_count_private_rooms  1     57556 168890340 250925
## + amenities                                     1     54342 168893553 250925
## + instant_bookable                              1     50294 168897601 250926
## + review_scores_value                           1     42976 168904920 250927
## + review_scores_checkin                         1     38233 168909662 250928
## + calculated_host_listings_count_entire_homes   1     27480 168920416 250930
## + host_total_listings_count                     1     26397 168921499 250930
## + host_since                                    1     13035 168934860 250932
## + review_scores_accuracy                        1     12596 168935299 250932
## <none>                                                      168947895 250932
## + first_review                                  1      7082 168940814 250933
## + review_scores_communication                   1      5171 168942724 250934
## + reviews_per_month                             1      2405 168945490 250934
## + cancellation_policy                           1      1937 168945958 250934
## + host_identity_verified                        1       187 168947708 250934
## + bed_type                                      4      8011 168939884 250939
## 
## Step:  AIC=250758.5
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30
## 
##                                                Df Sum of Sq       RSS    AIC
## + last_review                                   1    678405 167244824 250643
## + minimum_nights_avg_ntm                        1    480157 167443073 250678
## + review_scores_cleanliness                     1    421697 167501532 250688
## + review_scores_rating                          1    223886 167699343 250722
## + guests_included                               1    198172 167725057 250726
## + availability_365                              1    165976 167757254 250732
## + beds                                          1    124053 167799177 250739
## + host_is_superhost                             1     89793 167833436 250745
## + calculated_host_listings_count_shared_rooms   1     89222 167834007 250745
## + extra_people                                  1     79081 167844148 250747
## + is_location_exact                             1     75308 167847921 250747
## + calculated_host_listings_count_private_rooms  1     66555 167856675 250749
## + instant_bookable                              1     41121 167882108 250753
## + first_review                                  1     40430 167882799 250753
## + amenities                                     1     25547 167897682 250756
## + review_scores_checkin                         1     20590 167902639 250757
## + host_total_listings_count                     1     19490 167903739 250757
## + host_identity_verified                        1     16918 167906311 250758
## + review_scores_value                           1     11618 167911611 250758
## <none>                                                      167923229 250758
## + calculated_host_listings_count_entire_homes   1     11357 167911872 250759
## + host_since                                    1      1402 167921827 250760
## + review_scores_accuracy                        1      1161 167922068 250760
## + reviews_per_month                             1      1132 167922098 250760
## + cancellation_policy                           1       826 167922403 250760
## + review_scores_communication                   1        33 167923196 250760
## + bed_type                                      4     10174 167913056 250765
## 
## Step:  AIC=250643.4
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review
## 
##                                                Df Sum of Sq       RSS    AIC
## + minimum_nights_avg_ntm                        1    499024 166745801 250559
## + review_scores_cleanliness                     1    454869 166789955 250567
## + review_scores_rating                          1    233052 167011772 250605
## + availability_365                              1    222559 167022265 250607
## + guests_included                               1    204594 167040230 250610
## + host_is_superhost                             1    159737 167085087 250618
## + instant_bookable                              1    128359 167116465 250623
## + amenities                                     1    127737 167117087 250623
## + beds                                          1    121483 167123342 250624
## + is_location_exact                             1    100360 167144464 250628
## + extra_people                                  1     89435 167155389 250630
## + calculated_host_listings_count_shared_rooms   1     75743 167169081 250632
## + calculated_host_listings_count_private_rooms  1     45701 167199123 250637
## + first_review                                  1     26714 167218110 250641
## + review_scores_checkin                         1     23054 167221771 250641
## + host_total_listings_count                     1     20459 167224365 250642
## + host_since                                    1     17965 167226859 250642
## + calculated_host_listings_count_entire_homes   1     15712 167229112 250643
## + reviews_per_month                             1     12276 167232548 250643
## <none>                                                      167244824 250643
## + review_scores_value                           1     10371 167234453 250644
## + cancellation_policy                           1      1077 167243747 250645
## + review_scores_communication                   1       918 167243906 250645
## + review_scores_accuracy                        1       462 167244363 250645
## + host_identity_verified                        1       246 167244579 250645
## + bed_type                                      4     11317 167233507 250649
## 
## Step:  AIC=250558.9
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm
## 
##                                                Df Sum of Sq       RSS    AIC
## + review_scores_cleanliness                     1    421963 166323838 250488
## + availability_365                              1    299122 166446678 250509
## + review_scores_rating                          1    222566 166523235 250522
## + host_total_listings_count                     1    202611 166543190 250526
## + guests_included                               1    200404 166545397 250526
## + host_is_superhost                             1    163342 166582459 250533
## + amenities                                     1    150767 166595033 250535
## + instant_bookable                              1    145177 166600624 250536
## + beds                                          1    119088 166626713 250540
## + calculated_host_listings_count_entire_homes   1    118120 166627681 250540
## + is_location_exact                             1    109491 166636310 250542
## + calculated_host_listings_count_shared_rooms   1     69643 166676158 250549
## + extra_people                                  1     65802 166679998 250549
## + calculated_host_listings_count_private_rooms  1     30711 166715090 250556
## + first_review                                  1     27084 166718717 250556
## + review_scores_checkin                         1     26581 166719220 250556
## + host_since                                    1     18991 166726810 250558
## + review_scores_value                           1     18590 166727211 250558
## <none>                                                      166745801 250559
## + reviews_per_month                             1      7754 166738047 250560
## + review_scores_communication                   1      2379 166743421 250560
## + cancellation_policy                           1      1175 166744625 250561
## + review_scores_accuracy                        1       977 166744824 250561
## + host_identity_verified                        1       858 166744943 250561
## + bed_type                                      4     11250 166734550 250565
## 
## Step:  AIC=250487.6
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness
## 
##                                                Df Sum of Sq       RSS    AIC
## + review_scores_value                           1    362339 165961499 250427
## + availability_365                              1    313501 166010337 250435
## + review_scores_accuracy                        1    215575 166108263 250452
## + host_total_listings_count                     1    194605 166129233 250456
## + review_scores_checkin                         1    189644 166134194 250457
## + guests_included                               1    188533 166135305 250457
## + instant_bookable                              1    166970 166156868 250461
## + beds                                          1    114125 166209713 250470
## + review_scores_communication                   1    113318 166210520 250470
## + calculated_host_listings_count_entire_homes   1    106331 166217507 250471
## + is_location_exact                             1    105088 166218750 250471
## + amenities                                     1     97801 166226037 250473
## + host_is_superhost                             1     89764 166234073 250474
## + calculated_host_listings_count_shared_rooms   1     57844 166265994 250480
## + extra_people                                  1     57151 166266687 250480
## + first_review                                  1     33407 166290430 250484
## + host_since                                    1     26494 166297344 250485
## + calculated_host_listings_count_private_rooms  1     20689 166303149 250486
## <none>                                                      166323838 250488
## + reviews_per_month                             1      4693 166319145 250489
## + host_identity_verified                        1      3632 166320206 250489
## + review_scores_rating                          1      1851 166321986 250489
## + cancellation_policy                           1       914 166322924 250489
## + bed_type                                      4     11146 166312692 250494
## 
## Step:  AIC=250426.5
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value
## 
##                                                Df Sum of Sq       RSS    AIC
## + availability_365                              1    263555 165697944 250383
## + guests_included                               1    197531 165763968 250394
## + review_scores_rating                          1    183729 165777770 250396
## + host_total_listings_count                     1    178295 165783204 250397
## + instant_bookable                              1    156694 165804805 250401
## + host_is_superhost                             1    116246 165845253 250408
## + beds                                          1    113565 165847934 250409
## + amenities                                     1    105839 165855661 250410
## + is_location_exact                             1    101455 165860044 250411
## + calculated_host_listings_count_entire_homes   1     89683 165871816 250413
## + review_scores_checkin                         1     72134 165889365 250416
## + calculated_host_listings_count_shared_rooms   1     58983 165902516 250418
## + extra_people                                  1     55849 165905650 250419
## + review_scores_accuracy                        1     47259 165914240 250420
## + first_review                                  1     30743 165930756 250423
## + calculated_host_listings_count_private_rooms  1     27519 165933980 250424
## + host_since                                    1     18469 165943030 250425
## + review_scores_communication                   1     17678 165943821 250425
## <none>                                                      165961499 250427
## + reviews_per_month                             1      5006 165956493 250428
## + host_identity_verified                        1      1977 165959522 250428
## + cancellation_policy                           1        78 165961421 250428
## + bed_type                                      4     10376 165951123 250433
## 
## Step:  AIC=250382.5
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365
## 
##                                                Df Sum of Sq       RSS    AIC
## + guests_included                               1    196217 165501727 250350
## + review_scores_rating                          1    191084 165506860 250351
## + instant_bookable                              1    160687 165537257 250356
## + host_total_listings_count                     1    154065 165543880 250358
## + beds                                          1    122572 165575372 250363
## + host_is_superhost                             1    111289 165586656 250365
## + is_location_exact                             1    102888 165595056 250367
## + amenities                                     1     90769 165607175 250369
## + review_scores_checkin                         1     72395 165625550 250372
## + calculated_host_listings_count_shared_rooms   1     72160 165625784 250372
## + first_review                                  1     65259 165632685 250373
## + calculated_host_listings_count_entire_homes   1     55434 165642510 250375
## + extra_people                                  1     54624 165643320 250375
## + calculated_host_listings_count_private_rooms  1     50494 165647450 250376
## + review_scores_accuracy                        1     42024 165655920 250377
## + host_since                                    1     18976 165678968 250381
## + review_scores_communication                   1     14402 165683542 250382
## <none>                                                      165697944 250383
## + reviews_per_month                             1      6171 165691774 250383
## + host_identity_verified                        1      2031 165695913 250384
## + cancellation_policy                           1      1288 165696657 250384
## + bed_type                                      4     11056 165686889 250389
## 
## Step:  AIC=250350.2
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included
## 
##                                                Df Sum of Sq       RSS    AIC
## + review_scores_rating                          1    194944 165306783 250318
## + instant_bookable                              1    163799 165337928 250324
## + host_total_listings_count                     1    157691 165344037 250325
## + beds                                          1    156705 165345022 250325
## + host_is_superhost                             1    104340 165397387 250334
## + is_location_exact                             1    103345 165398382 250334
## + review_scores_checkin                         1     78615 165423112 250339
## + amenities                                     1     76558 165425170 250339
## + first_review                                  1     76086 165425641 250339
## + calculated_host_listings_count_shared_rooms   1     72077 165429650 250340
## + calculated_host_listings_count_private_rooms  1     52360 165449367 250343
## + calculated_host_listings_count_entire_homes   1     51156 165450572 250343
## + review_scores_accuracy                        1     44827 165456900 250344
## + host_since                                    1     19811 165481917 250349
## + review_scores_communication                   1     16653 165485074 250349
## + extra_people                                  1     13767 165487961 250350
## <none>                                                      165501727 250350
## + reviews_per_month                             1      5890 165495838 250351
## + cancellation_policy                           1      2600 165499127 250352
## + host_identity_verified                        1      2290 165499437 250352
## + bed_type                                      4     10604 165491124 250356
## 
## Step:  AIC=250318.1
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating
## 
##                                                Df Sum of Sq       RSS    AIC
## + review_scores_checkin                         1    189558 165117225 250287
## + review_scores_accuracy                        1    187988 165118795 250287
## + instant_bookable                              1    182048 165124735 250288
## + host_total_listings_count                     1    157071 165149712 250293
## + beds                                          1    149763 165157020 250294
## + review_scores_communication                   1    109757 165197026 250301
## + is_location_exact                             1    106980 165199803 250301
## + host_is_superhost                             1     94775 165212008 250304
## + first_review                                  1     86139 165220644 250305
## + calculated_host_listings_count_shared_rooms   1     71939 165234844 250308
## + amenities                                     1     65755 165241028 250309
## + calculated_host_listings_count_private_rooms  1     54829 165251954 250311
## + calculated_host_listings_count_entire_homes   1     51493 165255290 250311
## + host_since                                    1     30606 165276177 250315
## + extra_people                                  1     12472 165294311 250318
## <none>                                                      165306783 250318
## + reviews_per_month                             1      8930 165297853 250319
## + host_identity_verified                        1      4762 165302021 250319
## + cancellation_policy                           1      2258 165304526 250320
## + bed_type                                      4     10144 165296639 250324
## 
## Step:  AIC=250287
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin
## 
##                                                Df Sum of Sq       RSS    AIC
## + instant_bookable                              1    170898 164946327 250259
## + host_total_listings_count                     1    157048 164960177 250261
## + beds                                          1    149504 164967720 250263
## + review_scores_accuracy                        1    118308 164998916 250268
## + is_location_exact                             1    104518 165012706 250271
## + host_is_superhost                             1     98578 165018647 250272
## + calculated_host_listings_count_shared_rooms   1     71547 165045678 250276
## + amenities                                     1     69943 165047281 250277
## + first_review                                  1     65901 165051324 250277
## + calculated_host_listings_count_private_rooms  1     48400 165068825 250280
## + calculated_host_listings_count_entire_homes   1     48346 165068879 250280
## + review_scores_communication                   1     27276 165089949 250284
## + host_since                                    1     22361 165094864 250285
## + extra_people                                  1     12035 165105190 250287
## <none>                                                      165117225 250287
## + reviews_per_month                             1      6827 165110398 250288
## + host_identity_verified                        1      2788 165114437 250288
## + cancellation_policy                           1      1961 165115264 250289
## + bed_type                                      4     10238 165106986 250293
## 
## Step:  AIC=250259
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable
## 
##                                                Df Sum of Sq       RSS    AIC
## + beds                                          1    147082 164799244 250235
## + host_total_listings_count                     1    143453 164802874 250236
## + review_scores_accuracy                        1    114843 164831484 250241
## + host_is_superhost                             1    113423 164832904 250241
## + is_location_exact                             1     97254 164849073 250244
## + calculated_host_listings_count_shared_rooms   1     80913 164865414 250247
## + amenities                                     1     79199 164867128 250247
## + calculated_host_listings_count_private_rooms  1     43311 164903016 250253
## + calculated_host_listings_count_entire_homes   1     34526 164911801 250255
## + first_review                                  1     30463 164915863 250256
## + review_scores_communication                   1     25679 164920647 250256
## + extra_people                                  1     14808 164931519 250258
## <none>                                                      164946327 250259
## + host_since                                    1      3852 164942474 250260
## + cancellation_policy                           1       714 164945613 250261
## + reviews_per_month                             1       321 164946006 250261
## + host_identity_verified                        1       155 164946172 250261
## + bed_type                                      4      9917 164936410 250265
## 
## Step:  AIC=250235.2
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds
## 
##                                                Df Sum of Sq       RSS    AIC
## + host_total_listings_count                     1    141300 164657945 250212
## + review_scores_accuracy                        1    115300 164683944 250217
## + host_is_superhost                             1    115279 164683966 250217
## + is_location_exact                             1     97265 164701979 250220
## + amenities                                     1     86408 164712837 250222
## + calculated_host_listings_count_shared_rooms   1     65688 164733556 250226
## + calculated_host_listings_count_private_rooms  1     45022 164754222 250229
## + calculated_host_listings_count_entire_homes   1     35182 164764063 250231
## + first_review                                  1     31771 164767473 250232
## + review_scores_communication                   1     24423 164774821 250233
## + extra_people                                  1     15478 164783766 250234
## <none>                                                      164799244 250235
## + host_since                                    1      5519 164793726 250236
## + cancellation_policy                           1       548 164798697 250237
## + reviews_per_month                             1       332 164798913 250237
## + host_identity_verified                        1        27 164799217 250237
## + bed_type                                      4      9522 164789723 250242
## 
## Step:  AIC=250212.4
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count
## 
##                                                Df Sum of Sq       RSS    AIC
## + review_scores_accuracy                        1    117230 164540714 250194
## + host_is_superhost                             1    106630 164551314 250196
## + is_location_exact                             1     90011 164567934 250199
## + amenities                                     1     76655 164581290 250201
## + calculated_host_listings_count_shared_rooms   1     67577 164590368 250202
## + calculated_host_listings_count_private_rooms  1     59434 164598511 250204
## + first_review                                  1     22553 164635392 250210
## + review_scores_communication                   1     21670 164636275 250211
## + extra_people                                  1     21505 164636440 250211
## <none>                                                      164657945 250212
## + host_since                                    1      3050 164654895 250214
## + cancellation_policy                           1       462 164657483 250214
## + host_identity_verified                        1       394 164657551 250214
## + reviews_per_month                             1       256 164657689 250214
## + calculated_host_listings_count_entire_homes   1         8 164657936 250214
## + bed_type                                      4      9267 164648678 250219
## 
## Step:  AIC=250193.8
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy
## 
##                                                Df Sum of Sq       RSS    AIC
## + host_is_superhost                             1    110169 164430545 250176
## + is_location_exact                             1     88087 164452628 250180
## + amenities                                     1     78322 164462392 250182
## + calculated_host_listings_count_shared_rooms   1     68584 164472130 250184
## + calculated_host_listings_count_private_rooms  1     59081 164481633 250185
## + extra_people                                  1     21293 164519422 250192
## + first_review                                  1     18370 164522345 250193
## + review_scores_communication                   1     11451 164529264 250194
## <none>                                                      164540714 250194
## + host_since                                    1      2023 164538692 250195
## + host_identity_verified                        1       554 164540161 250196
## + cancellation_policy                           1       342 164540372 250196
## + reviews_per_month                             1        74 164540641 250196
## + calculated_host_listings_count_entire_homes   1        69 164540645 250196
## + bed_type                                      4      8836 164531879 250200
## 
## Step:  AIC=250176.4
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost
## 
##                                                Df Sum of Sq       RSS    AIC
## + is_location_exact                             1     90830 164339715 250162
## + calculated_host_listings_count_shared_rooms   1     65722 164364823 250167
## + calculated_host_listings_count_private_rooms  1     54732 164375813 250169
## + amenities                                     1     51301 164379244 250169
## + first_review                                  1     27446 164403099 250174
## + extra_people                                  1     18871 164411674 250175
## + review_scores_communication                   1     12193 164418352 250176
## <none>                                                      164430545 250176
## + host_since                                    1      4730 164425815 250178
## + cancellation_policy                           1       852 164429693 250178
## + reviews_per_month                             1       689 164429856 250178
## + calculated_host_listings_count_entire_homes   1       224 164430321 250178
## + host_identity_verified                        1        13 164430532 250178
## + bed_type                                      4      9607 164420938 250183
## 
## Step:  AIC=250162.4
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact
## 
##                                                Df Sum of Sq       RSS    AIC
## + calculated_host_listings_count_shared_rooms   1     64072 164275644 250153
## + calculated_host_listings_count_private_rooms  1     50290 164289425 250156
## + amenities                                     1     47129 164292587 250156
## + first_review                                  1     19628 164320087 250161
## + extra_people                                  1     19462 164320254 250161
## + review_scores_communication                   1     12743 164326973 250162
## <none>                                                      164339715 250162
## + host_since                                    1      3056 164336659 250164
## + cancellation_policy                           1       729 164338986 250164
## + reviews_per_month                             1       671 164339044 250164
## + calculated_host_listings_count_entire_homes   1       651 164339065 250164
## + host_identity_verified                        1        32 164339683 250164
## + bed_type                                      4      9668 164330048 250169
## 
## Step:  AIC=250153.1
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms
## 
##                                                Df Sum of Sq       RSS    AIC
## + amenities                                     1     48950 164226694 250147
## + calculated_host_listings_count_private_rooms  1     48094 164227549 250147
## + first_review                                  1     22396 164253248 250151
## + extra_people                                  1     21227 164254417 250151
## + review_scores_communication                   1     11884 164263760 250153
## <none>                                                      164275644 250153
## + host_since                                    1      3847 164271797 250154
## + calculated_host_listings_count_entire_homes   1       948 164274696 250155
## + reviews_per_month                             1       762 164274882 250155
## + cancellation_policy                           1       598 164275046 250155
## + host_identity_verified                        1         1 164275643 250155
## + bed_type                                      4      9794 164265850 250159
## 
## Step:  AIC=250146.5
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms + 
##     amenities
## 
##                                                Df Sum of Sq       RSS    AIC
## + calculated_host_listings_count_private_rooms  1     46557 164180136 250140
## + first_review                                  1     32371 164194323 250143
## + extra_people                                  1     17197 164209497 250145
## + review_scores_communication                   1     12838 164213856 250146
## <none>                                                      164226694 250147
## + host_since                                    1      7043 164219650 250147
## + calculated_host_listings_count_entire_homes   1      1602 164225092 250148
## + cancellation_policy                           1      1567 164225126 250148
## + reviews_per_month                             1      1020 164225674 250148
## + host_identity_verified                        1       253 164226441 250148
## + bed_type                                      4      9863 164216831 250153
## 
## Step:  AIC=250140.3
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms + 
##     amenities + calculated_host_listings_count_private_rooms
## 
##                                               Df Sum of Sq       RSS    AIC
## + first_review                                 1     36314 164143822 250136
## + extra_people                                 1     15833 164164304 250140
## + review_scores_communication                  1     13488 164166648 250140
## <none>                                                     164180136 250140
## + host_since                                   1      8643 164171493 250141
## + calculated_host_listings_count_entire_homes  1      2389 164177747 250142
## + cancellation_policy                          1      1250 164178886 250142
## + reviews_per_month                            1       822 164179314 250142
## + host_identity_verified                       1       606 164179530 250142
## + bed_type                                     4     10070 164170066 250147
## 
## Step:  AIC=250135.9
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms + 
##     amenities + calculated_host_listings_count_private_rooms + 
##     first_review
## 
##                                               Df Sum of Sq       RSS    AIC
## + extra_people                                 1   18638.1 164125184 250135
## <none>                                                     164143822 250136
## + review_scores_communication                  1   11346.6 164132475 250136
## + calculated_host_listings_count_entire_homes  1    3547.2 164140275 250137
## + host_identity_verified                       1     888.9 164142933 250138
## + reviews_per_month                            1     779.7 164143042 250138
## + cancellation_policy                          1     178.8 164143643 250138
## + host_since                                   1       1.4 164143820 250138
## + bed_type                                     4    8601.4 164135220 250142
## 
## Step:  AIC=250134.6
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms + 
##     amenities + calculated_host_listings_count_private_rooms + 
##     first_review + extra_people
## 
##                                               Df Sum of Sq       RSS    AIC
## + review_scores_communication                  1   11530.9 164113653 250135
## <none>                                                     164125184 250135
## + calculated_host_listings_count_entire_homes  1    2621.2 164122562 250136
## + host_identity_verified                       1     922.6 164124261 250136
## + reviews_per_month                            1     781.9 164124402 250136
## + cancellation_policy                          1     424.3 164124759 250137
## + host_since                                   1       0.3 164125183 250137
## + bed_type                                     4    8578.0 164116606 250141
## 
## Step:  AIC=250134.6
## price ~ accommodates + neighbourhood_group_cleansed + cleaning_fee + 
##     room_type + bathrooms + bedrooms + property_type + review_scores_location + 
##     number_of_reviews_ltm + availability_30 + last_review + minimum_nights_avg_ntm + 
##     review_scores_cleanliness + review_scores_value + availability_365 + 
##     guests_included + review_scores_rating + review_scores_checkin + 
##     instant_bookable + beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms + 
##     amenities + calculated_host_listings_count_private_rooms + 
##     first_review + extra_people + review_scores_communication
## 
##                                               Df Sum of Sq       RSS    AIC
## <none>                                                     164113653 250135
## + calculated_host_listings_count_entire_homes  1    2597.0 164111056 250136
## + host_identity_verified                       1     956.2 164112697 250136
## + reviews_per_month                            1     746.2 164112907 250136
## + cancellation_policy                          1     421.7 164113231 250137
## + host_since                                   1       2.9 164113650 250137
## + bed_type                                     4    8587.0 164105066 250141

summary(forward_selection)

## 
## Call:
## lm(formula = price ~ accommodates + neighbourhood_group_cleansed + 
##     cleaning_fee + room_type + bathrooms + bedrooms + property_type + 
##     review_scores_location + number_of_reviews_ltm + availability_30 + 
##     last_review + minimum_nights_avg_ntm + review_scores_cleanliness + 
##     review_scores_value + availability_365 + guests_included + 
##     review_scores_rating + review_scores_checkin + instant_bookable + 
##     beds + host_total_listings_count + review_scores_accuracy + 
##     host_is_superhost + is_location_exact + calculated_host_listings_count_shared_rooms + 
##     amenities + calculated_host_listings_count_private_rooms + 
##     first_review + extra_people + review_scores_communication, 
##     data = train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -456.86  -37.07   -6.50   23.49  836.68 
## 
## Coefficients:
##                                                Estimate Std. Error t value
## (Intercept)                                  -1.223e+02  7.751e+00 -15.775
## accommodates                                  1.298e+01  4.415e-01  29.405
## neighbourhood_group_cleansedBrooklyn          1.982e+01  2.893e+00   6.850
## neighbourhood_group_cleansedManhattan         6.688e+01  2.935e+00  22.787
## neighbourhood_group_cleansedQueens            9.231e+00  3.050e+00   3.027
## neighbourhood_group_cleansedStaten Island    -1.962e+01  5.627e+00  -3.487
## cleaning_fee                                  3.112e-01  1.139e-02  27.330
## room_typeHotel room                           2.070e+00  6.552e+00   0.316
## room_typePrivate room                        -5.119e+01  1.176e+00 -43.543
## room_typeShared room                         -6.830e+01  3.676e+00 -18.582
## bathrooms                                     3.167e+01  1.217e+00  26.034
## bedrooms                                      2.187e+01  8.735e-01  25.031
## property_typeHotel                            3.134e+01  3.583e+00   8.748
## property_typeHouse                           -2.174e+00  1.493e+00  -1.456
## property_typeCondominium                      2.527e+01  2.481e+00  10.186
## property_typeLoft                             4.199e+01  2.680e+00  15.667
## property_typeOther                            2.644e+01  7.770e+00   3.403
## review_scores_location                        1.268e+01  6.970e-01  18.188
## number_of_reviews_ltm                        -3.876e-01  3.157e-02 -12.279
## availability_30                               4.561e-01  4.938e-02   9.237
## last_review                                   1.809e-02  1.363e-03  13.275
## minimum_nights_avg_ntm                       -1.973e-01  1.743e-02 -11.315
## review_scores_cleanliness                     3.942e+00  6.011e-01   6.558
## review_scores_value                          -6.125e+00  7.915e-01  -7.739
## availability_365                              3.196e-02  4.423e-03   7.227
## guests_included                               2.826e+00  4.823e-01   5.859
## review_scores_rating                          8.651e-01  1.028e-01   8.418
## review_scores_checkin                        -2.708e+00  8.183e-01  -3.310
## instant_bookable                              4.690e+00  9.682e-01   4.844
## beds                                         -3.113e+00  6.268e-01  -4.966
## host_total_listings_count                     3.419e-02  7.241e-03   4.722
## review_scores_accuracy                       -3.574e+00  8.332e-01  -4.289
## host_is_superhost                             4.738e+00  1.215e+00   3.899
## is_location_exact                            -4.189e+00  1.181e+00  -3.546
## calculated_host_listings_count_shared_rooms  -1.901e+00  5.488e-01  -3.464
## amenities                                     1.768e-01  5.711e-02   3.096
## calculated_host_listings_count_private_rooms -2.151e-01  7.280e-02  -2.956
## first_review                                 -1.911e-03  7.500e-04  -2.548
## extra_people                                  3.520e-02  1.934e-02   1.820
## review_scores_communication                  -1.228e+00  8.616e-01  -1.425
##                                              Pr(>|t|)    
## (Intercept)                                   < 2e-16 ***
## accommodates                                  < 2e-16 ***
## neighbourhood_group_cleansedBrooklyn         7.52e-12 ***
## neighbourhood_group_cleansedManhattan         < 2e-16 ***
## neighbourhood_group_cleansedQueens           0.002472 ** 
## neighbourhood_group_cleansedStaten Island    0.000490 ***
## cleaning_fee                                  < 2e-16 ***
## room_typeHotel room                          0.752057    
## room_typePrivate room                         < 2e-16 ***
## room_typeShared room                          < 2e-16 ***
## bathrooms                                     < 2e-16 ***
## bedrooms                                      < 2e-16 ***
## property_typeHotel                            < 2e-16 ***
## property_typeHouse                           0.145275    
## property_typeCondominium                      < 2e-16 ***
## property_typeLoft                             < 2e-16 ***
## property_typeOther                           0.000668 ***
## review_scores_location                        < 2e-16 ***
## number_of_reviews_ltm                         < 2e-16 ***
## availability_30                               < 2e-16 ***
## last_review                                   < 2e-16 ***
## minimum_nights_avg_ntm                        < 2e-16 ***
## review_scores_cleanliness                    5.55e-11 ***
## review_scores_value                          1.04e-14 ***
## availability_365                             5.07e-13 ***
## guests_included                              4.71e-09 ***
## review_scores_rating                          < 2e-16 ***
## review_scores_checkin                        0.000935 ***
## instant_bookable                             1.28e-06 ***
## beds                                         6.86e-07 ***
## host_total_listings_count                    2.35e-06 ***
## review_scores_accuracy                       1.80e-05 ***
## host_is_superhost                            9.67e-05 ***
## is_location_exact                            0.000391 ***
## calculated_host_listings_count_shared_rooms  0.000532 ***
## amenities                                    0.001966 ** 
## calculated_host_listings_count_private_rooms 0.003124 ** 
## first_review                                 0.010824 *  
## extra_people                                 0.068727 .  
## review_scores_communication                  0.154246    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 75.37 on 28890 degrees of freedom
## Multiple R-squared:  0.5308, Adjusted R-squared:  0.5302 
## F-statistic: 838.1 on 39 and 28890 DF,  p-value: < 2.2e-16

# predict using test
pred_fw_test = predict(forward_selection, newdata = test)
rmse_fw_test = sqrt(mean((pred_fw_test - test$price)^2)); rmse_fw_test

## [1] 78.31976

Lasso

Also, we can use lasso to see if we can see different results.

library(glmnet)

## Loading required package: Matrix

## 
## Attaching package: 'Matrix'

## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack

## Loaded glmnet 4.1-2

## 
## Attaching package: 'glmnet'

## The following object is masked from 'package:na.tools':
## 
##     na.replace

x = model.matrix(price ~ .-1, data = train)
y = train$price

cv_lasso = cv.glmnet(x = x, 
                     y = y, 
                     alpha = 1,
                     type.measure = 'mse')
plot(cv_lasso)

coef(cv_lasso, s = cv_lasso$lambda.1se) %>%
  round(4)

## 50 x 1 sparse Matrix of class "dgCMatrix"
##                                                    s1
## (Intercept)                                  -85.7050
## cleaning_fee                                   0.3426
## beds                                           .     
## host_since                                     .     
## host_is_superhost                              .     
## host_total_listings_count                      .     
## host_identity_verified                         .     
## first_review                                   .     
## last_review                                    0.0073
## reviews_per_month                              .     
## neighbourhood_group_cleansedBronx             -6.9862
## neighbourhood_group_cleansedBrooklyn           .     
## neighbourhood_group_cleansedManhattan         46.9042
## neighbourhood_group_cleansedQueens            -4.7274
## neighbourhood_group_cleansedStaten Island    -16.0757
## is_location_exact                              .     
## property_typeHotel                            22.0941
## property_typeHouse                            -0.9176
## property_typeCondominium                      15.9769
## property_typeLoft                             31.4390
## property_typeOther                             .     
## room_typeHotel room                            .     
## room_typePrivate room                        -47.1655
## room_typeShared room                         -57.2408
## accommodates                                  13.0065
## bathrooms                                     26.7691
## bedrooms                                      18.3798
## bed_typeCouch                                  .     
## bed_typeFuton                                  .     
## bed_typePull-out Sofa                          .     
## bed_typeReal Bed                               .     
## amenities                                      .     
## guests_included                                1.6709
## extra_people                                   .     
## minimum_nights_avg_ntm                        -0.0667
## availability_30                                0.2567
## availability_365                               0.0156
## number_of_reviews_ltm                         -0.2385
## review_scores_rating                           .     
## review_scores_accuracy                         .     
## review_scores_cleanliness                      2.1383
## review_scores_checkin                          .     
## review_scores_communication                    .     
## review_scores_location                         8.8555
## review_scores_value                            .     
## instant_bookable                               .     
## cancellation_policy                            .     
## calculated_host_listings_count_entire_homes    .     
## calculated_host_listings_count_private_rooms   .     
## calculated_host_listings_count_shared_rooms   -0.6297

Performance of Lasso

We use features selected from Lasso to run a linear model to see the results.

R2: 0.5261
RSE: 75.73
RMSE_test: 78.84

# subset variables
analysis_la <- analysis %>% select(-beds, -host_since, -host_is_superhost, -host_identity_verified,
                                   -host_total_listings_count, -first_review, -reviews_per_month,
                                   -is_location_exact, -bed_type, -amenities, -extra_people, 
                                   -review_scores_rating, -review_scores_accuracy, 
                                   -review_scores_checkin, -review_scores_communication,
                                   -review_scores_value, -cancellation_policy, -instant_bookable,
                                   -calculated_host_listings_count_entire_homes, 
                                   -calculated_host_listings_count_private_rooms)
train_la <- train %>% select(names(analysis_la))
test_la <- test %>% select(names(analysis_la))
# Lasso model
mod_lasso <- lm(price ~ ., data = train_la)
summary(mod_lasso)

## 
## Call:
## lm(formula = price ~ ., data = train_la)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -419.35  -37.41   -6.43   23.08  840.23 
## 
## Coefficients:
##                                               Estimate Std. Error t value
## (Intercept)                                 -1.461e+02  6.827e+00 -21.403
## cleaning_fee                                 3.331e-01  1.112e-02  29.948
## last_review                                  1.384e-02  1.176e-03  11.772
## neighbourhood_group_cleansedBrooklyn         1.874e+01  2.901e+00   6.460
## neighbourhood_group_cleansedManhattan        6.702e+01  2.940e+00  22.796
## neighbourhood_group_cleansedQueens           8.999e+00  3.058e+00   2.942
## neighbourhood_group_cleansedStaten Island   -1.969e+01  5.652e+00  -3.484
## property_typeHotel                           3.330e+01  3.589e+00   9.277
## property_typeHouse                          -3.016e+00  1.492e+00  -2.021
## property_typeCondominium                     2.627e+01  2.483e+00  10.582
## property_typeLoft                            4.249e+01  2.684e+00  15.829
## property_typeOther                           2.728e+01  7.804e+00   3.495
## room_typeHotel room                          7.240e+00  6.550e+00   1.105
## room_typePrivate room                       -5.082e+01  1.160e+00 -43.812
## room_typeShared room                        -6.730e+01  3.672e+00 -18.330
## accommodates                                 1.236e+01  3.943e-01  31.338
## bathrooms                                    3.114e+01  1.215e+00  25.634
## bedrooms                                     2.010e+01  8.441e-01  23.814
## guests_included                              2.651e+00  4.633e-01   5.721
## minimum_nights_avg_ntm                      -1.561e-01  1.617e-02  -9.651
## availability_30                              4.837e-01  4.920e-02   9.832
## availability_365                             3.271e-02  4.331e-03   7.552
## number_of_reviews_ltm                       -3.659e-01  2.949e-02 -12.409
## review_scores_cleanliness                    3.676e+00  4.339e-01   8.471
## review_scores_location                       1.056e+01  6.302e-01  16.763
## calculated_host_listings_count_shared_rooms -1.945e+00  5.484e-01  -3.546
##                                             Pr(>|t|)    
## (Intercept)                                  < 2e-16 ***
## cleaning_fee                                 < 2e-16 ***
## last_review                                  < 2e-16 ***
## neighbourhood_group_cleansedBrooklyn        1.07e-10 ***
## neighbourhood_group_cleansedManhattan        < 2e-16 ***
## neighbourhood_group_cleansedQueens          0.003260 ** 
## neighbourhood_group_cleansedStaten Island   0.000495 ***
## property_typeHotel                           < 2e-16 ***
## property_typeHouse                          0.043282 *  
## property_typeCondominium                     < 2e-16 ***
## property_typeLoft                            < 2e-16 ***
## property_typeOther                          0.000474 ***
## room_typeHotel room                         0.269018    
## room_typePrivate room                        < 2e-16 ***
## room_typeShared room                         < 2e-16 ***
## accommodates                                 < 2e-16 ***
## bathrooms                                    < 2e-16 ***
## bedrooms                                     < 2e-16 ***
## guests_included                             1.07e-08 ***
## minimum_nights_avg_ntm                       < 2e-16 ***
## availability_30                              < 2e-16 ***
## availability_365                            4.41e-14 ***
## number_of_reviews_ltm                        < 2e-16 ***
## review_scores_cleanliness                    < 2e-16 ***
## review_scores_location                       < 2e-16 ***
## calculated_host_listings_count_shared_rooms 0.000392 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 75.73 on 28904 degrees of freedom
## Multiple R-squared:  0.5261, Adjusted R-squared:  0.5257 
## F-statistic:  1284 on 25 and 28904 DF,  p-value: < 2.2e-16

# check the results
pred_la_test = predict(mod_lasso, newdata = test_la)
rmse_la_test = sqrt(mean((pred_la_test - test_la$price)^2)); rmse_la_test

## [1] 78.83772

Tune Lasso

Lambda = 0.0515 R2: 0.5225
RMSE_cv: 76.71

trControl_la <-trainControl(method = "cv", number = 5)
tuneGrid_la <- expand.grid(alpha = 1, lambda = seq(0.001, 0.1, by = 0.0005))

mod_lasso_cv <- train(price ~ ., data = analysis_la,
                method = 'glmnet', 
                trControl = trControl_la, 
                tuneGrid = tuneGrid_la)

mod_lasso_cv$bestTune

##     alpha lambda
## 102     1 0.0515

min(mod_lasso_cv$results$Rsquared)

## [1] 0.5228462

rmse_la_cv = min(mod_lasso_cv$results$RMSE); rmse_la_cv

## [1] 76.71351

Regression Tree

Default Tree

RMSE_train: 80.99
RMSE_test: 84.47

library(rpart); library(rpart.plot)
tree1 = rpart(price ~ ., data = train, method = "anova")
rpart.plot(tree1)

# importance of variables
tree1$variable.importance

##                                    room_type 
##                                   87761272.6 
## calculated_host_listings_count_private_rooms 
##                                   77947260.6 
##  calculated_host_listings_count_entire_homes 
##                                   77578612.5 
##                                 accommodates 
##                                   49911988.3 
##                                 cleaning_fee 
##                                   46263446.0 
##                                    bathrooms 
##                                   42840660.7 
##                                         beds 
##                                   34224636.1 
##                 neighbourhood_group_cleansed 
##                                   18464042.1 
##                                     bedrooms 
##                                   15890704.6 
##                              guests_included 
##                                    3611633.2 
##                                property_type 
##                                    2560865.1 
##                       review_scores_location 
##                                    1740243.2 
##                                 extra_people 
##                                    1453064.0 
##                             instant_bookable 
##                                    1193089.4 
##                                   host_since 
##                                     911086.5

# predict using tree1
pred_t1_train = predict(tree1, data = train)
rmse_t1_train = sqrt(mean((pred_t1_train - train$price)^2)); rmse_t1_train

## [1] 80.99496

pred_t1 = predict(tree1, newdata = test)

# calculate rmse_test
library(Metrics)

## 
## Attaching package: 'Metrics'

## The following objects are masked from 'package:caret':
## 
##     precision, recall

rmse_t1_test = rmse(actual = test$price, predicted = pred_t1); rmse_t1_test

## [1] 84.46786

Tune the Tree

cp: 0.0006
R2: 0.5420
RMSE_cv: 74.41

tuneGrid_t = expand.grid(cp = seq(0, 0.1, 0.0001))
trControl_t = trainControl(method = 'cv', number = 5)

tree_cv = train(price ~ ., data = analysis, method = 'rpart', 
                trControl = trControl_t, tuneGrid = tuneGrid_t)

# results
tree_cv$results

##          cp     RMSE  Rsquared      MAE    RMSESD  RsquaredSD     MAESD
## 1    0.0000 78.35870 0.5240736 46.25648 2.5865586 0.023369227 1.0503146
## 2    0.0001 75.89232 0.5438725 43.49768 2.3996985 0.022060487 0.8687960
## 3    0.0002 75.34151 0.5470605 43.29352 2.6911057 0.025128613 0.9469221
## 4    0.0003 75.03711 0.5486948 43.37928 2.9141199 0.027164514 1.0177955
## 5    0.0004 75.00310 0.5481510 43.51832 3.0397944 0.027606217 1.1053426
## 6    0.0005 74.84726 0.5491196 43.69734 2.7495651 0.024612510 0.9713130
## 7    0.0006 75.02897 0.5462774 43.91723 2.7315213 0.023715455 0.9311744
## 8    0.0007 74.89090 0.5469524 44.07361 2.9617661 0.026816662 1.0817705
## 9    0.0008 75.13022 0.5436261 44.25630 2.8938299 0.026299220 1.0802950
## 10   0.0009 75.30321 0.5412609 44.40404 2.6711328 0.023052364 0.9888520
## 11   0.0010 75.25728 0.5415612 44.43282 2.5070217 0.021653181 0.9541850
## 12   0.0011 75.41992 0.5393620 44.50189 2.4514844 0.021417172 0.9620841
## 13   0.0012 75.62972 0.5367295 44.67468 2.2513852 0.019019881 0.9310828
## 14   0.0013 75.65597 0.5363131 44.74666 2.2226731 0.021604534 1.0051093
## 15   0.0014 75.66442 0.5361481 44.77969 2.1803739 0.021332623 0.9289719
## 16   0.0015 75.80638 0.5342308 44.86095 2.0536848 0.020003694 0.8658489
## 17   0.0016 75.93858 0.5326528 44.89349 1.9073220 0.017509549 0.8255074
## 18   0.0017 76.10708 0.5305904 44.96333 1.8174850 0.017691806 0.7722281
## 19   0.0018 76.16273 0.5298853 45.00294 1.7575667 0.016351513 0.7510039
## 20   0.0019 76.24087 0.5289044 45.04892 1.9730204 0.017070755 0.8040714
## 21   0.0020 76.31501 0.5278728 45.15618 2.0118646 0.017218857 0.8542861
## 22   0.0021 76.51218 0.5254078 45.37825 1.9201676 0.016460301 0.7912197
## 23   0.0022 76.56300 0.5247740 45.43046 1.9524317 0.016703479 0.8167814
## 24   0.0023 76.56300 0.5247740 45.43046 1.9524317 0.016703479 0.8167814
## 25   0.0024 76.69092 0.5231639 45.49615 1.7157034 0.015027921 0.7755873
## 26   0.0025 76.69092 0.5231639 45.49615 1.7157034 0.015027921 0.7755873
## 27   0.0026 76.82617 0.5214952 45.54204 1.7597002 0.015231131 0.8012986
## 28   0.0027 76.92170 0.5203139 45.61443 1.9053879 0.016579286 0.8456049
## 29   0.0028 77.03171 0.5189655 45.70227 1.9358010 0.015953379 0.8264038
## 30   0.0029 77.16018 0.5173383 45.81648 1.9434551 0.016001192 0.8155856
## 31   0.0030 77.12285 0.5177910 45.81293 1.8805002 0.015768002 0.8115140
## 32   0.0031 77.15849 0.5173734 45.83042 1.8328864 0.014876015 0.7754742
## 33   0.0032 77.15849 0.5173734 45.83042 1.8328864 0.014876015 0.7754742
## 34   0.0033 77.23333 0.5164064 45.88219 1.7667781 0.015940874 0.8062490
## 35   0.0034 77.35128 0.5148600 45.99014 1.7066680 0.016845788 0.8388701
## 36   0.0035 77.38883 0.5143940 46.02279 1.7715147 0.016967928 0.8586523
## 37   0.0036 77.70798 0.5104001 46.18505 2.0613003 0.019204309 0.9658254
## 38   0.0037 78.06218 0.5060229 46.30735 1.6142148 0.009859843 0.7192150
## 39   0.0038 78.06218 0.5060229 46.30735 1.6142148 0.009859843 0.7192150
## 40   0.0039 78.06218 0.5060229 46.30735 1.6142148 0.009859843 0.7192150
## 41   0.0040 78.06218 0.5060229 46.30735 1.6142148 0.009859843 0.7192150
## 42   0.0041 78.12661 0.5051730 46.38217 1.5344791 0.010928747 0.7744030
## 43   0.0042 78.09569 0.5055204 46.38238 1.5715984 0.010463288 0.7746028
## 44   0.0043 78.06990 0.5057838 46.36170 1.6041850 0.010135965 0.7560285
## 45   0.0044 78.06990 0.5057838 46.36170 1.6041850 0.010135965 0.7560285
## 46   0.0045 78.06990 0.5057838 46.36170 1.6041850 0.010135965 0.7560285
## 47   0.0046 78.06990 0.5057838 46.36170 1.6041850 0.010135965 0.7560285
## 48   0.0047 78.06990 0.5057838 46.36170 1.6041850 0.010135965 0.7560285
## 49   0.0048 78.42044 0.5012896 46.50487 1.6485589 0.012715887 0.8067963
## 50   0.0049 78.46304 0.5007723 46.51350 1.6916229 0.013227186 0.8098593
## 51   0.0050 78.46304 0.5007723 46.51350 1.6916229 0.013227186 0.8098593
## 52   0.0051 78.46304 0.5007723 46.51350 1.6916229 0.013227186 0.8098593
## 53   0.0052 78.46304 0.5007723 46.51350 1.6916229 0.013227186 0.8098593
## 54   0.0053 78.46304 0.5007723 46.51350 1.6916229 0.013227186 0.8098593
## 55   0.0054 78.62203 0.4986213 46.60215 1.5350562 0.016005912 0.9100523
## 56   0.0055 78.62203 0.4986213 46.60215 1.5350562 0.016005912 0.9100523
## 57   0.0056 78.83318 0.4959629 46.69581 1.7444529 0.017341885 0.9958390
## 58   0.0057 79.03183 0.4934203 46.75903 1.6933334 0.017270076 0.9867561
## 59   0.0058 79.50257 0.4873318 46.91924 2.0383705 0.021467682 1.0527903
## 60   0.0059 79.50257 0.4873318 46.91924 2.0383705 0.021467682 1.0527903
## 61   0.0060 79.60593 0.4859990 46.96042 2.0762394 0.021507666 1.0490376
## 62   0.0061 79.63869 0.4855666 46.98338 2.0238448 0.020603023 0.9995222
## 63   0.0062 79.82533 0.4832327 47.07719 2.1914430 0.021165971 1.0419447
## 64   0.0063 80.04627 0.4803684 47.29303 2.1249987 0.020490343 0.7679483
## 65   0.0064 80.13870 0.4791687 47.40396 2.1845507 0.020989311 0.7905623
## 66   0.0065 80.13870 0.4791687 47.40396 2.1845507 0.020989311 0.7905623
## 67   0.0066 80.30819 0.4769602 47.53703 1.9725143 0.020704580 0.8332090
## 68   0.0067 80.46769 0.4749890 47.59203 1.7362304 0.016712060 0.7142045
## 69   0.0068 80.54466 0.4739932 47.62318 1.6378695 0.014784974 0.6474451
## 70   0.0069 80.71386 0.4718111 47.94690 1.8006887 0.015270915 0.8899786
## 71   0.0070 80.73197 0.4715258 47.96501 1.8169919 0.015494602 0.9144923
## 72   0.0071 80.73197 0.4715258 47.96501 1.8169919 0.015494602 0.9144923
## 73   0.0072 80.92307 0.4690126 48.05087 2.0307018 0.018326453 1.0443600
## 74   0.0073 81.21482 0.4652127 48.11719 1.7685203 0.012167458 0.9266121
## 75   0.0074 81.21482 0.4652127 48.11719 1.7685203 0.012167458 0.9266121
## 76   0.0075 81.21482 0.4652127 48.11719 1.7685203 0.012167458 0.9266121
## 77   0.0076 81.27078 0.4644545 48.16212 1.6670704 0.012169711 0.9389691
## 78   0.0077 81.43910 0.4622657 48.25753 1.6025283 0.010065273 0.7836915
## 79   0.0078 81.43910 0.4622657 48.25753 1.6025283 0.010065273 0.7836915
## 80   0.0079 81.50480 0.4614394 48.29665 1.6649766 0.008767957 0.7919687
## 81   0.0080 81.50480 0.4614394 48.29665 1.6649766 0.008767957 0.7919687
## 82   0.0081 81.59862 0.4601312 48.36033 1.4843109 0.008979323 0.8203945
## 83   0.0082 81.59862 0.4601312 48.36033 1.4843109 0.008979323 0.8203945
## 84   0.0083 81.59862 0.4601312 48.36033 1.4843109 0.008979323 0.8203945
## 85   0.0084 81.59862 0.4601312 48.36033 1.4843109 0.008979323 0.8203945
## 86   0.0085 81.59862 0.4601312 48.36033 1.4843109 0.008979323 0.8203945
## 87   0.0086 81.78054 0.4577074 48.45921 1.5174615 0.011224849 0.7610853
## 88   0.0087 81.78054 0.4577074 48.45921 1.5174615 0.011224849 0.7610853
## 89   0.0088 81.78054 0.4577074 48.45921 1.5174615 0.011224849 0.7610853
## 90   0.0089 81.94261 0.4555619 48.68950 1.4929455 0.010108319 0.4388342
## 91   0.0090 81.94261 0.4555619 48.68950 1.4929455 0.010108319 0.4388342
## 92   0.0091 81.94261 0.4555619 48.68950 1.4929455 0.010108319 0.4388342
## 93   0.0092 81.94261 0.4555619 48.68950 1.4929455 0.010108319 0.4388342
## 94   0.0093 82.32148 0.4504666 49.01411 1.7767598 0.015769394 0.5470620
## 95   0.0094 82.32148 0.4504666 49.01411 1.7767598 0.015769394 0.5470620
## 96   0.0095 82.32148 0.4504666 49.01411 1.7767598 0.015769394 0.5470620
## 97   0.0096 82.66691 0.4459273 49.33675 1.9201004 0.012854572 0.3594243
## 98   0.0097 82.66691 0.4459273 49.33675 1.9201004 0.012854572 0.3594243
## 99   0.0098 82.69979 0.4454522 49.37964 1.9473543 0.013840555 0.4291570
## 100  0.0099 82.69979 0.4454522 49.37964 1.9473543 0.013840555 0.4291570
## 101  0.0100 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 102  0.0101 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 103  0.0102 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 104  0.0103 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 105  0.0104 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 106  0.0105 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 107  0.0106 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 108  0.0107 83.14974 0.4394669 49.85986 2.2754555 0.012594652 0.8398792
## 109  0.0108 83.36355 0.4364836 50.14087 1.8386994 0.008779273 0.5818083
## 110  0.0109 83.36355 0.4364836 50.14087 1.8386994 0.008779273 0.5818083
## 111  0.0110 83.36355 0.4364836 50.14087 1.8386994 0.008779273 0.5818083
## 112  0.0111 83.36355 0.4364836 50.14087 1.8386994 0.008779273 0.5818083
## 113  0.0112 83.54077 0.4340922 50.33097 1.8300502 0.007675000 0.3385661
## 114  0.0113 83.70944 0.4318068 50.51692 1.9324513 0.009288662 0.5553869
## 115  0.0114 83.70944 0.4318068 50.51692 1.9324513 0.009288662 0.5553869
## 116  0.0115 83.87113 0.4296751 50.66130 2.1218998 0.007934946 0.7373507
## 117  0.0116 83.87113 0.4296751 50.66130 2.1218998 0.007934946 0.7373507
## 118  0.0117 83.87113 0.4296751 50.66130 2.1218998 0.007934946 0.7373507
## 119  0.0118 83.87113 0.4296751 50.66130 2.1218998 0.007934946 0.7373507
## 120  0.0119 83.87113 0.4296751 50.66130 2.1218998 0.007934946 0.7373507
## 121  0.0120 84.30246 0.4237989 50.91813 2.3271180 0.012694381 0.8656404
## 122  0.0121 84.30246 0.4237989 50.91813 2.3271180 0.012694381 0.8656404
## 123  0.0122 84.30246 0.4237989 50.91813 2.3271180 0.012694381 0.8656404
## 124  0.0123 84.44106 0.4218803 50.97663 2.3287766 0.013657618 0.7930975
## 125  0.0124 84.44106 0.4218803 50.97663 2.3287766 0.013657618 0.7930975
## 126  0.0125 84.44106 0.4218803 50.97663 2.3287766 0.013657618 0.7930975
## 127  0.0126 84.44106 0.4218803 50.97663 2.3287766 0.013657618 0.7930975
## 128  0.0127 84.44106 0.4218803 50.97663 2.3287766 0.013657618 0.7930975
## 129  0.0128 84.60772 0.4195541 51.06634 2.3847121 0.016330292 0.7147956
## 130  0.0129 84.60772 0.4195541 51.06634 2.3847121 0.016330292 0.7147956
## 131  0.0130 84.60772 0.4195541 51.06634 2.3847121 0.016330292 0.7147956
## 132  0.0131 84.76671 0.4174415 51.15395 2.5306704 0.015328101 0.8164398
## 133  0.0132 84.76671 0.4174415 51.15395 2.5306704 0.015328101 0.8164398
## 134  0.0133 84.81970 0.4167075 51.18696 2.4173979 0.013913617 0.7692213
## 135  0.0134 84.81970 0.4167075 51.18696 2.4173979 0.013913617 0.7692213
## 136  0.0135 84.81970 0.4167075 51.18696 2.4173979 0.013913617 0.7692213
## 137  0.0136 84.81970 0.4167075 51.18696 2.4173979 0.013913617 0.7692213
## 138  0.0137 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 139  0.0138 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 140  0.0139 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 141  0.0140 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 142  0.0141 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 143  0.0142 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 144  0.0143 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 145  0.0144 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 146  0.0145 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 147  0.0146 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 148  0.0147 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 149  0.0148 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 150  0.0149 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 151  0.0150 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 152  0.0151 85.01085 0.4140220 51.35422 2.5272321 0.018283787 0.7321577
## 153  0.0152 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 154  0.0153 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 155  0.0154 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 156  0.0155 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 157  0.0156 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 158  0.0157 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 159  0.0158 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 160  0.0159 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 161  0.0160 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 162  0.0161 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 163  0.0162 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 164  0.0163 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 165  0.0164 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 166  0.0165 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 167  0.0166 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 168  0.0167 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 169  0.0168 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 170  0.0169 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 171  0.0170 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 172  0.0171 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 173  0.0172 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 174  0.0173 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 175  0.0174 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 176  0.0175 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 177  0.0176 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 178  0.0177 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 179  0.0178 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 180  0.0179 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 181  0.0180 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 182  0.0181 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 183  0.0182 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 184  0.0183 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 185  0.0184 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 186  0.0185 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 187  0.0186 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 188  0.0187 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 189  0.0188 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 190  0.0189 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 191  0.0190 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 192  0.0191 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 193  0.0192 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 194  0.0193 85.25006 0.4106144 51.64028 2.7534052 0.024832741 1.0436826
## 195  0.0194 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 196  0.0195 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 197  0.0196 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 198  0.0197 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 199  0.0198 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 200  0.0199 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 201  0.0200 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 202  0.0201 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 203  0.0202 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 204  0.0203 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 205  0.0204 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 206  0.0205 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 207  0.0206 85.54038 0.4065375 51.94176 2.1670338 0.020564013 0.7536422
## 208  0.0207 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 209  0.0208 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 210  0.0209 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 211  0.0210 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 212  0.0211 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 213  0.0212 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 214  0.0213 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 215  0.0214 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 216  0.0215 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 217  0.0216 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 218  0.0217 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 219  0.0218 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 220  0.0219 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 221  0.0220 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 222  0.0221 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 223  0.0222 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 224  0.0223 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 225  0.0224 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 226  0.0225 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 227  0.0226 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 228  0.0227 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 229  0.0228 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 230  0.0229 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 231  0.0230 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 232  0.0231 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 233  0.0232 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 234  0.0233 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 235  0.0234 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 236  0.0235 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 237  0.0236 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 238  0.0237 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 239  0.0238 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 240  0.0239 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 241  0.0240 85.80947 0.4026657 52.14559 1.6657016 0.019817064 0.8547046
## 242  0.0241 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 243  0.0242 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 244  0.0243 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 245  0.0244 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 246  0.0245 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 247  0.0246 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 248  0.0247 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 249  0.0248 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 250  0.0249 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 251  0.0250 86.12294 0.3981159 52.29635 1.2038414 0.023427662 1.0573699
## 252  0.0251 86.47093 0.3932886 52.54187 0.9943377 0.020726793 0.7346598
## 253  0.0252 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 254  0.0253 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 255  0.0254 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 256  0.0255 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 257  0.0256 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 258  0.0257 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 259  0.0258 86.82477 0.3885121 52.74821 1.4089546 0.015296363 0.7025695
## 260  0.0259 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 261  0.0260 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 262  0.0261 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 263  0.0262 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 264  0.0263 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 265  0.0264 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 266  0.0265 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 267  0.0266 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 268  0.0267 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 269  0.0268 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 270  0.0269 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 271  0.0270 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 272  0.0271 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 273  0.0272 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 274  0.0273 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 275  0.0274 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 276  0.0275 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 277  0.0276 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 278  0.0277 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 279  0.0278 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 280  0.0279 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 281  0.0280 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 282  0.0281 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 283  0.0282 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 284  0.0283 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 285  0.0284 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 286  0.0285 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 287  0.0286 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 288  0.0287 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 289  0.0288 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 290  0.0289 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 291  0.0290 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 292  0.0291 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 293  0.0292 87.12848 0.3842689 52.94653 1.3612025 0.010946072 0.5503209
## 294  0.0293 87.65205 0.3770634 53.13429 2.2861920 0.013415145 0.7333301
## 295  0.0294 87.65205 0.3770634 53.13429 2.2861920 0.013415145 0.7333301
## 296  0.0295 87.65205 0.3770634 53.13429 2.2861920 0.013415145 0.7333301
## 297  0.0296 87.65205 0.3770634 53.13429 2.2861920 0.013415145 0.7333301
## 298  0.0297 87.65205 0.3770634 53.13429 2.2861920 0.013415145 0.7333301
## 299  0.0298 87.65205 0.3770634 53.13429 2.2861920 0.013415145 0.7333301
## 300  0.0299 88.12535 0.3702926 53.29669 2.5364951 0.022752360 0.7806552
## 301  0.0300 88.12535 0.3702926 53.29669 2.5364951 0.022752360 0.7806552
## 302  0.0301 88.12535 0.3702926 53.29669 2.5364951 0.022752360 0.7806552
## 303  0.0302 88.59944 0.3635584 53.75099 2.4071188 0.018800166 0.5212583
## 304  0.0303 88.59944 0.3635584 53.75099 2.4071188 0.018800166 0.5212583
## 305  0.0304 88.59944 0.3635584 53.75099 2.4071188 0.018800166 0.5212583
## 306  0.0305 88.59944 0.3635584 53.75099 2.4071188 0.018800166 0.5212583
## 307  0.0306 88.97154 0.3580192 54.07673 2.8062697 0.029620901 0.9167824
## 308  0.0307 88.97154 0.3580192 54.07673 2.8062697 0.029620901 0.9167824
## 309  0.0308 88.97154 0.3580192 54.07673 2.8062697 0.029620901 0.9167824
## 310  0.0309 88.97154 0.3580192 54.07673 2.8062697 0.029620901 0.9167824
## 311  0.0310 89.39693 0.3519703 54.48062 3.3513329 0.032261053 1.2605641
## 312  0.0311 89.39693 0.3519703 54.48062 3.3513329 0.032261053 1.2605641
## 313  0.0312 89.39693 0.3519703 54.48062 3.3513329 0.032261053 1.2605641
## 314  0.0313 89.79930 0.3462412 54.64421 3.1929320 0.026639987 1.0489556
## 315  0.0314 89.79930 0.3462412 54.64421 3.1929320 0.026639987 1.0489556
## 316  0.0315 89.79930 0.3462412 54.64421 3.1929320 0.026639987 1.0489556
## 317  0.0316 89.79930 0.3462412 54.64421 3.1929320 0.026639987 1.0489556
## 318  0.0317 89.79930 0.3462412 54.64421 3.1929320 0.026639987 1.0489556
## 319  0.0318 89.79930 0.3462412 54.64421 3.1929320 0.026639987 1.0489556
## 320  0.0319 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 321  0.0320 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 322  0.0321 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 323  0.0322 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 324  0.0323 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 325  0.0324 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 326  0.0325 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 327  0.0326 90.17013 0.3408655 55.01508 3.2669666 0.026410321 0.9916569
## 328  0.0327 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 329  0.0328 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 330  0.0329 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 331  0.0330 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 332  0.0331 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 333  0.0332 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 334  0.0333 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 335  0.0334 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 336  0.0335 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 337  0.0336 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 338  0.0337 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 339  0.0338 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 340  0.0339 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 341  0.0340 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 342  0.0341 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 343  0.0342 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 344  0.0343 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 345  0.0344 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 346  0.0345 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 347  0.0346 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 348  0.0347 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 349  0.0348 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 350  0.0349 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 351  0.0350 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 352  0.0351 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 353  0.0352 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 354  0.0353 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 355  0.0354 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 356  0.0355 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 357  0.0356 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 358  0.0357 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 359  0.0358 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 360  0.0359 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 361  0.0360 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 362  0.0361 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 363  0.0362 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 364  0.0363 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 365  0.0364 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 366  0.0365 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 367  0.0366 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 368  0.0367 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 369  0.0368 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 370  0.0369 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 371  0.0370 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 372  0.0371 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 373  0.0372 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 374  0.0373 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 375  0.0374 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 376  0.0375 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 377  0.0376 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 378  0.0377 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 379  0.0378 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 380  0.0379 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 381  0.0380 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 382  0.0381 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 383  0.0382 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 384  0.0383 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 385  0.0384 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 386  0.0385 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 387  0.0386 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 388  0.0387 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 389  0.0388 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 390  0.0389 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 391  0.0390 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 392  0.0391 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 393  0.0392 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 394  0.0393 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 395  0.0394 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 396  0.0395 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 397  0.0396 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 398  0.0397 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 399  0.0398 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 400  0.0399 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 401  0.0400 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 402  0.0401 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 403  0.0402 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 404  0.0403 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 405  0.0404 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 406  0.0405 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 407  0.0406 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 408  0.0407 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 409  0.0408 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 410  0.0409 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 411  0.0410 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 412  0.0411 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 413  0.0412 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 414  0.0413 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 415  0.0414 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 416  0.0415 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 417  0.0416 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 418  0.0417 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 419  0.0418 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 420  0.0419 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 421  0.0420 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 422  0.0421 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 423  0.0422 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 424  0.0423 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 425  0.0424 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 426  0.0425 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 427  0.0426 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 428  0.0427 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 429  0.0428 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 430  0.0429 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 431  0.0430 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 432  0.0431 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 433  0.0432 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 434  0.0433 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 435  0.0434 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 436  0.0435 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 437  0.0436 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 438  0.0437 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 439  0.0438 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 440  0.0439 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 441  0.0440 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 442  0.0441 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 443  0.0442 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 444  0.0443 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 445  0.0444 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 446  0.0445 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 447  0.0446 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 448  0.0447 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 449  0.0448 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 450  0.0449 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 451  0.0450 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 452  0.0451 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 453  0.0452 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 454  0.0453 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 455  0.0454 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 456  0.0455 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 457  0.0456 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 458  0.0457 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 459  0.0458 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 460  0.0459 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 461  0.0460 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 462  0.0461 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 463  0.0462 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 464  0.0463 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 465  0.0464 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 466  0.0465 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 467  0.0466 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 468  0.0467 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 469  0.0468 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 470  0.0469 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 471  0.0470 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 472  0.0471 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 473  0.0472 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 474  0.0473 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 475  0.0474 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 476  0.0475 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 477  0.0476 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 478  0.0477 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 479  0.0478 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 480  0.0479 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 481  0.0480 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 482  0.0481 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 483  0.0482 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 484  0.0483 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 485  0.0484 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 486  0.0485 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 487  0.0486 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 488  0.0487 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 489  0.0488 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 490  0.0489 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 491  0.0490 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 492  0.0491 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 493  0.0492 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 494  0.0493 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 495  0.0494 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 496  0.0495 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 497  0.0496 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 498  0.0497 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 499  0.0498 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 500  0.0499 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 501  0.0500 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 502  0.0501 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 503  0.0502 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 504  0.0503 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 505  0.0504 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 506  0.0505 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 507  0.0506 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 508  0.0507 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 509  0.0508 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 510  0.0509 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 511  0.0510 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 512  0.0511 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 513  0.0512 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 514  0.0513 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 515  0.0514 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 516  0.0515 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 517  0.0516 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 518  0.0517 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 519  0.0518 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 520  0.0519 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 521  0.0520 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 522  0.0521 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 523  0.0522 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 524  0.0523 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 525  0.0524 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 526  0.0525 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 527  0.0526 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 528  0.0527 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 529  0.0528 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 530  0.0529 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 531  0.0530 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 532  0.0531 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 533  0.0532 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 534  0.0533 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 535  0.0534 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 536  0.0535 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 537  0.0536 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 538  0.0537 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 539  0.0538 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 540  0.0539 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 541  0.0540 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 542  0.0541 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 543  0.0542 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 544  0.0543 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 545  0.0544 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 546  0.0545 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 547  0.0546 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 548  0.0547 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 549  0.0548 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 550  0.0549 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 551  0.0550 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 552  0.0551 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 553  0.0552 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 554  0.0553 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 555  0.0554 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 556  0.0555 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 557  0.0556 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 558  0.0557 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 559  0.0558 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 560  0.0559 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 561  0.0560 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 562  0.0561 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 563  0.0562 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 564  0.0563 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 565  0.0564 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 566  0.0565 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 567  0.0566 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 568  0.0567 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 569  0.0568 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 570  0.0569 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 571  0.0570 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 572  0.0571 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 573  0.0572 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 574  0.0573 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 575  0.0574 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 576  0.0575 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 577  0.0576 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 578  0.0577 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 579  0.0578 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 580  0.0579 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 581  0.0580 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 582  0.0581 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 583  0.0582 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 584  0.0583 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 585  0.0584 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 586  0.0585 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 587  0.0586 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 588  0.0587 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 589  0.0588 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 590  0.0589 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 591  0.0590 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 592  0.0591 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 593  0.0592 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 594  0.0593 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 595  0.0594 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 596  0.0595 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 597  0.0596 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 598  0.0597 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 599  0.0598 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 600  0.0599 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 601  0.0600 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 602  0.0601 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 603  0.0602 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 604  0.0603 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 605  0.0604 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 606  0.0605 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 607  0.0606 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 608  0.0607 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 609  0.0608 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 610  0.0609 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 611  0.0610 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 612  0.0611 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 613  0.0612 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 614  0.0613 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 615  0.0614 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 616  0.0615 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 617  0.0616 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 618  0.0617 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 619  0.0618 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 620  0.0619 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 621  0.0620 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 622  0.0621 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 623  0.0622 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 624  0.0623 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 625  0.0624 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 626  0.0625 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 627  0.0626 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 628  0.0627 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 629  0.0628 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 630  0.0629 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 631  0.0630 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 632  0.0631 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 633  0.0632 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 634  0.0633 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 635  0.0634 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 636  0.0635 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 637  0.0636 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 638  0.0637 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 639  0.0638 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 640  0.0639 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 641  0.0640 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 642  0.0641 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 643  0.0642 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 644  0.0643 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 645  0.0644 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 646  0.0645 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 647  0.0646 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 648  0.0647 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 649  0.0648 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 650  0.0649 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 651  0.0650 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 652  0.0651 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 653  0.0652 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 654  0.0653 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 655  0.0654 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 656  0.0655 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 657  0.0656 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 658  0.0657 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 659  0.0658 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 660  0.0659 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 661  0.0660 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 662  0.0661 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 663  0.0662 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 664  0.0663 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 665  0.0664 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 666  0.0665 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 667  0.0666 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 668  0.0667 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 669  0.0668 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 670  0.0669 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 671  0.0670 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 672  0.0671 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 673  0.0672 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 674  0.0673 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 675  0.0674 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 676  0.0675 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 677  0.0676 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 678  0.0677 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 679  0.0678 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 680  0.0679 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 681  0.0680 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 682  0.0681 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 683  0.0682 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 684  0.0683 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 685  0.0684 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 686  0.0685 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 687  0.0686 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 688  0.0687 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 689  0.0688 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 690  0.0689 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 691  0.0690 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 692  0.0691 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 693  0.0692 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 694  0.0693 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 695  0.0694 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 696  0.0695 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 697  0.0696 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 698  0.0697 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 699  0.0698 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 700  0.0699 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 701  0.0700 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 702  0.0701 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 703  0.0702 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 704  0.0703 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 705  0.0704 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 706  0.0705 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 707  0.0706 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 708  0.0707 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 709  0.0708 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 710  0.0709 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 711  0.0710 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 712  0.0711 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 713  0.0712 90.48645 0.3363377 55.08117 3.2022368 0.024190479 0.9472876
## 714  0.0713 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 715  0.0714 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 716  0.0715 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 717  0.0716 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 718  0.0717 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 719  0.0718 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 720  0.0719 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 721  0.0720 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 722  0.0721 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 723  0.0722 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 724  0.0723 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 725  0.0724 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 726  0.0725 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 727  0.0726 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 728  0.0727 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 729  0.0728 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 730  0.0729 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 731  0.0730 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 732  0.0731 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 733  0.0732 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 734  0.0733 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 735  0.0734 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 736  0.0735 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 737  0.0736 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 738  0.0737 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 739  0.0738 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 740  0.0739 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 741  0.0740 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 742  0.0741 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 743  0.0742 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 744  0.0743 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 745  0.0744 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 746  0.0745 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 747  0.0746 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 748  0.0747 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 749  0.0748 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 750  0.0749 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 751  0.0750 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 752  0.0751 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 753  0.0752 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 754  0.0753 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 755  0.0754 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 756  0.0755 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 757  0.0756 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 758  0.0757 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 759  0.0758 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 760  0.0759 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 761  0.0760 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 762  0.0761 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 763  0.0762 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 764  0.0763 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 765  0.0764 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 766  0.0765 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 767  0.0766 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 768  0.0767 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 769  0.0768 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 770  0.0769 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 771  0.0770 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 772  0.0771 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 773  0.0772 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 774  0.0773 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 775  0.0774 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 776  0.0775 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 777  0.0776 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 778  0.0777 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 779  0.0778 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 780  0.0779 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 781  0.0780 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 782  0.0781 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 783  0.0782 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 784  0.0783 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 785  0.0784 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 786  0.0785 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 787  0.0786 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 788  0.0787 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 789  0.0788 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 790  0.0789 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 791  0.0790 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 792  0.0791 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 793  0.0792 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 794  0.0793 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 795  0.0794 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 796  0.0795 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 797  0.0796 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 798  0.0797 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 799  0.0798 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 800  0.0799 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 801  0.0800 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 802  0.0801 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 803  0.0802 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 804  0.0803 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 805  0.0804 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 806  0.0805 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 807  0.0806 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 808  0.0807 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 809  0.0808 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 810  0.0809 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 811  0.0810 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 812  0.0811 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 813  0.0812 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 814  0.0813 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 815  0.0814 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 816  0.0815 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 817  0.0816 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 818  0.0817 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 819  0.0818 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 820  0.0819 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 821  0.0820 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 822  0.0821 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 823  0.0822 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 824  0.0823 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 825  0.0824 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 826  0.0825 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 827  0.0826 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 828  0.0827 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 829  0.0828 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 830  0.0829 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 831  0.0830 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 832  0.0831 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 833  0.0832 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 834  0.0833 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 835  0.0834 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 836  0.0835 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 837  0.0836 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 838  0.0837 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 839  0.0838 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 840  0.0839 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 841  0.0840 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 842  0.0841 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 843  0.0842 91.31372 0.3238630 55.49014 1.5654415 0.012186168 0.4958379
## 844  0.0843 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 845  0.0844 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 846  0.0845 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 847  0.0846 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 848  0.0847 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 849  0.0848 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 850  0.0849 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 851  0.0850 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 852  0.0851 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 853  0.0852 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 854  0.0853 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 855  0.0854 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 856  0.0855 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 857  0.0856 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 858  0.0857 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 859  0.0858 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 860  0.0859 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 861  0.0860 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 862  0.0861 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 863  0.0862 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 864  0.0863 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 865  0.0864 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 866  0.0865 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 867  0.0866 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 868  0.0867 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 869  0.0868 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 870  0.0869 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 871  0.0870 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 872  0.0871 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 873  0.0872 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 874  0.0873 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 875  0.0874 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 876  0.0875 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 877  0.0876 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 878  0.0877 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 879  0.0878 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 880  0.0879 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 881  0.0880 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 882  0.0881 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 883  0.0882 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 884  0.0883 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 885  0.0884 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 886  0.0885 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 887  0.0886 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 888  0.0887 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 889  0.0888 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 890  0.0889 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 891  0.0890 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 892  0.0891 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 893  0.0892 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 894  0.0893 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 895  0.0894 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 896  0.0895 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 897  0.0896 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 898  0.0897 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 899  0.0898 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 900  0.0899 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 901  0.0900 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 902  0.0901 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 903  0.0902 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 904  0.0903 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 905  0.0904 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 906  0.0905 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 907  0.0906 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 908  0.0907 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 909  0.0908 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 910  0.0909 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 911  0.0910 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 912  0.0911 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 913  0.0912 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 914  0.0913 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 915  0.0914 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 916  0.0915 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 917  0.0916 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 918  0.0917 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 919  0.0918 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 920  0.0919 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 921  0.0920 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 922  0.0921 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 923  0.0922 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 924  0.0923 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 925  0.0924 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 926  0.0925 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 927  0.0926 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 928  0.0927 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 929  0.0928 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 930  0.0929 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 931  0.0930 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 932  0.0931 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 933  0.0932 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 934  0.0933 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 935  0.0934 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 936  0.0935 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 937  0.0936 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 938  0.0937 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 939  0.0938 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 940  0.0939 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 941  0.0940 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 942  0.0941 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 943  0.0942 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 944  0.0943 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 945  0.0944 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 946  0.0945 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 947  0.0946 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 948  0.0947 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 949  0.0948 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 950  0.0949 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 951  0.0950 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 952  0.0951 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 953  0.0952 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 954  0.0953 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 955  0.0954 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 956  0.0955 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 957  0.0956 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 958  0.0957 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 959  0.0958 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 960  0.0959 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 961  0.0960 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 962  0.0961 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 963  0.0962 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 964  0.0963 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 965  0.0964 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 966  0.0965 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 967  0.0966 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 968  0.0967 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 969  0.0968 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 970  0.0969 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 971  0.0970 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 972  0.0971 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 973  0.0972 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 974  0.0973 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 975  0.0974 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 976  0.0975 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 977  0.0976 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 978  0.0977 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 979  0.0978 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 980  0.0979 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 981  0.0980 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 982  0.0981 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 983  0.0982 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 984  0.0983 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 985  0.0984 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 986  0.0985 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 987  0.0986 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 988  0.0987 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 989  0.0988 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 990  0.0989 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 991  0.0990 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 992  0.0991 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 993  0.0992 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 994  0.0993 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 995  0.0994 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 996  0.0995 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 997  0.0996 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 998  0.0997 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 999  0.0998 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 1000 0.0999 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541
## 1001 0.1000 92.51801 0.3046925 57.31628 1.8230095 0.050106968 4.2124541

plot(tree_cv)

tree_cv$bestTune

##      cp
## 6 5e-04

rmse_t1_cv = min(tree_cv$results$RMSE); rmse_t1_cv

## [1] 74.84726

Random Forest

Since tuning random forest is really time consuming, here I just show the codes random forest and results.

RMSE_train: 65.22
RMSE_test: 67.84

library(randomForest)
forest = randomForest(price ~ ., data = train, ntree = 1000)
varImpPlot(forest, n.var = nrow(forest$importance))

# predict by train
pred_rf_train = predict(forest)
rmse_rf_train = sqrt(mean((pred_rf_train - train$price)^2)); rmse_rf_train
# predict by test
pred_rf_test = predict(forest, newdata = test)
rmse_rf_test = sqrt(mean((pred_rf_test - test$price)^2)); rmse_rf_test

Ranger

Now, let’s try ranger to build the model. Considered of timing consumption, I simply put the code and the results here and did not run it.

RMSE_train: 30.65
RMSE_test: 68.61

library(ranger)
forest_ranger = ranger(price ~ ., data = train, num.trees = 2000)

# predict using train
pred_ranger_train = predict(forest_ranger, data = train, num.trees = 2000)
rmse_ranger_train = sqrt(mean((pred_ranger_train$predictions - train$price)^2)); rmse_ranger_train

# predict using test
pred_ranger_test = predict(forest_ranger, data = test, num.trees = 2000)
rmse_ranger_test = sqrt(mean((pred_ranger_test$predictions - test$price)^2)); rmse_ranger_test

Xgboost

Create Xgb.DMatrix

First, let’s convert data for xgboost model.

One-hot Encoding

library(vtreat)

## Loading required package: wrapr

## 
## Attaching package: 'wrapr'

## The following objects are masked from 'package:Matrix':
## 
##     pack, unpack

## The following object is masked from 'package:data.table':
## 
##     :=

## The following object is masked from 'package:dplyr':
## 
##     coalesce

## The following objects are masked from 'package:tidyr':
## 
##     pack, unpack

## The following object is masked from 'package:tibble':
## 
##     view

trt = designTreatmentsZ(dframe = analysis,
                        varlist = names(analysis)[2:(ncol(analysis)-1)])

## [1] "vtreat 1.6.3 inspecting inputs Tue Nov 30 11:53:20 2021"
## [1] "designing treatments Tue Nov 30 11:53:20 2021"
## [1] " have initial level statistics Tue Nov 30 11:53:20 2021"
## [1] " scoring treatments Tue Nov 30 11:53:20 2021"
## [1] "have treatment plan Tue Nov 30 11:53:20 2021"

newvars = trt$scoreFrame[trt$scoreFrame$code %in% c('clean','lev'),'varName']

# create analysis_dummy
price = analysis$price
analysis_dmy = prepare(treatmentplan = trt, dframe = analysis, varRestriction = newvars)
combined = cbind(price, analysis_dmy)
dim(combined)

## [1] 41322    52

# dummy scoring data
scoring_dmy = prepare(treatmentplan = trt, dframe = scoring, varRestriction = newvars)
id = scoring$id
dim(scoring_dmy)

## [1] 10333    51

Create xgb.DMatrix

library(xgboost)

## 
## Attaching package: 'xgboost'

## The following object is masked from 'package:dplyr':
## 
##     slice

# separate input and output for combined
combined_input = combined[, -1]
d_combined <- xgb.DMatrix(data = as.matrix(combined_input), label = price)
dim(d_combined)

## [1] 41322    51

# create scoring xgb.DMatrix
dscoring = xgb.DMatrix(data = as.matrix(scoring_dmy))
dim(dscoring)

## [1] 10333    51

Find the best hyperparameter values

Here, I simply put the code here and did not run it as it’s really time consuming. The best parameters are shown below.

# lambda and alpha for normalization
# max_depth and n_estimator for overfitting

trControl_xgb = trControl_la <-trainControl(method = "cv", number = 5)
xgb_grid = expand.grid(nrounds = 1500,
                       eta = c(0.1, 0.05, 0.01),
                       max_depth = c(8, 9, 12),
                       gamma = 0,
                       colsample_bytree = 0.8,
                       min_child_weight = c(2, 3, 4),
                       subsample = 0.8)

xgb_caret <- train(x = as.matrix(combined_input), 
                   y = combined$price, 
                   method = 'xgbTree', 
                   trControl= trControl_xgb, 
                   tuneGrid = xgb_grid)
xgb_caret$bestTune

Find the Best Iteration

RMSE_train: 33.46
RMSE_test: 62.83

# setting params
best_param <- as.list(xgb_caret$bestTune)
# cross validation using best_param
xgbcv <- xgb.cv(params = best_param, 
                nrounds = 1500, 
                data = d_combined, 
                nfold = 5, 
                showsd = T, 
                stratified = T, 
                print_every_n = 100, 
                early_stopping_rounds = 100, 
                maximize = F)
rmse_xgb_test = min(xgbcv$evaluation_log$test_rmse_mean); rmse_xgb_test

Build the Best Xgboost Model

mod_xgb <- xgb.train(params = default_param,
                     data = d_combined,
                     nrounds = xgbcv$best_iteration)

Results

Take a look at the result, we can see that xgboost generated the best RMSE, so I used xgboost model on scoring data. The scores on Kaggle are:
Public Leaderboard: 68.39
Private Leaderboard: 61.22

results <- data.frame(model = c("Linear Regression", "Forward Selection", "Lasso", "Tuned Lasso",
                                "Tree", "Tuned Tree", "Random Forest", "Ranger", "Xgboost"),
                      RMSE = round(c(rmse_mod1_test, rmse_fw_test, rmse_la_test, rmse_la_cv, 
                                     rmse_t1_test, rmse_t1_cv, 67.84, 68.61, 62.83), 4))
datatable(results)